goldrei.classic set theory for guided independent study

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Page 1: Goldrei.classic Set Theory for Guided Independent Study
Page 2: Goldrei.classic Set Theory for Guided Independent Study
Page 3: Goldrei.classic Set Theory for Guided Independent Study
Page 4: Goldrei.classic Set Theory for Guided Independent Study
Page 5: Goldrei.classic Set Theory for Guided Independent Study
Page 6: Goldrei.classic Set Theory for Guided Independent Study

� � �

* � * � 2 *

* * � 2 � � � �

* � 2 * 2 *

� * � * 2 � * 2 E * *

* * 2 � * � 2 2 �

� � 2 2 2 �

� � � * � P 2 � 2

� * � � * * 2

� * 2 * 2 2 � * � �

� * * * 2 * 2 * � 2

2 � � * 2 * 2 � *

E * * � * � 2 2 * *

� 2 2 � 2 � * * �

� * 2 2 � * � � � 2 2

* � * 2 � * E * 2 *

2 � * � * �

� � � 2

� E * 2 2 � P � *

E * © � � 2 2 * © © � 2 * *

2 2 P 2 � � 2 P * * E 2

� � 2 * * © * 2 � � � 2

E * 2 � * 2 2 * � P * �

� 2 � � * 2 � 2 * 2

* � 2 * 2 2 * *

� * © � 2 � � 2 * � � 2

� * � 2 � * © * * 2 � �

� 2 P * * � * * 2 � �

� E * * � 2 � � * � * � 2

� * * � 2 � * � * � 2 * * �

� 2 2 * © © 2 � * P � 2 � � *

* * � © 2 E * * 2

� 2 * � � * * � * * * 2 *

� * 2 2 2 * 2 2 * *

E � 2 � 2 2

� � � * * * � 2

* * � E � * � �

* � * 2 E 2 � P * * ©

2 2 * � � 2 * E 2

* © � P © * 2 � © 2 �

Page 7: Goldrei.classic Set Theory for Guided Independent Study

� � � � � �

� � � � �

� 2 � � � � �

2 � 2 � � � � � � � � �

� �

Page 8: Goldrei.classic Set Theory for Guided Independent Study

� & & & � . � � 3 � 3 & 3 � .

& 3 � = > > & & . = D 3 & D > �

. � D � � . .

& & O . � 3 � � � & D & � .

. � & & & � . � & & �

= � > D & � . & � & . � O . D 3

. 3 � 3 � . � � > � . &

. . = . . � . � � �

& & . & . D � � . D >

& 3 D � . D D . � . > 3 �

D . � = � 3 � � . . D � � = & > . D

& > � . . D � & � = D & D � > 3 D = D 3 � &

D . 3 3 � � = D � > > 3 =

D . 3 3 D 3 > � = 3 D � = 3 & � 3 � � > = > &

D & > O 3 � = � > O � > > O D � = = >

. O O . 3 & . & D & O & & 3 � = D & > . D

& . O D > � O � = > & & � 3 & D & Ì � O > � =

& � > � Ì = & > > O & D . � = � > �

& & . Ì & > & O . O . & & � = � 3 = &

& > O & � > & � & 3 � > � Ì � > D � � O

& . & � & > & .

. . D � � � � D 3 . � > D D Ì >

& > Ì D = = D D > & > . Ì � = D > Ì � Ì . D 3 D � =

D 3 & > . Ì . Ì D � > D � > � > . � . D 3 D � =

D & � D � > D 3 D O � > Ì 3 & 3 > � D &

> O � D D D � > � > = D � & Ì O D

� > D > 3 � & & > � D D > � = D 3 &

� 3 & > > 3 D = & D Ì . � > . D 3 & Ì 3 3 =

� � = & & � > . � & D D � = D O � & D O D D � & & > > � >

D & & . 3 D � & 3 D . 3

. � = = & � 3 & Ì � � � & � . � � = � > =

� & . � & O � . 3 3 D � = D 3 � � D 3 � > Ì . D 3 D =

D � 3 D & O � & D D O D > � � Ì D = � � & D & Ì > > D

. > O D = = � 3 D = � D > � > O D = = D =

> D & � = & D > = D & 3 Ì > & Ì D � = 3 Ì D � >

> > D D D � = > � D � � D � . D � O � = O

> O D D O & > 3 � O D O � D D D O > 3 D � = � > � � D

. & 3 > O D & D & . D 3 D � = D & � > 3 3

3 D & & D & > & & & &

. D D & & D & D D 3 & = D � � D Ì 3 D � O � > > O D & D � �

� � = . > D � > . D & D > 3 D Ì > & 3 O > & � D Ì

Page 9: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � % % � * % � * . � � % � �

� � 9 9 � 9 � % � � > 9 � � 9 �

� % 9 � . � � % 9 9 � 9 9 � � > � 9 �

� � 9 � � � � � � 9 � 9 � * �

U � U 9 9 � 9 � U � 9 � �

� % 9 U � � % 9 U � � % � > � 9 � U % � > �

9 � 9 � � % U � � � � �

> 9 U � 9 � U � > � * � � 9 � % > * � � * � �

� U 9 � U � > � * � > � � � � U 9 � 9 � 9 *

* � � 9 % � � . % � * � * � � � . � 9 %

� � U . 9 � � . * % � 9 � � � > * * � * � *

� > 9 * � � % � � � � % > * � * � � 9 � � � *

U U � � � U 9 9 9 * % � % 9 � * * � U % % . � � � � � � % . .

� % � 9 � � % � � 9 9 � 9 � U � 9 .

� U � � � � > 9 9 9 � � 9 > > � 9 * � % * % . � � 9 �

U 9 > � � � % *

� > 9 . 9 > * � % % � > � > � . � �

� * > % 9 9 � 9 � > 9 � � % * U � � � � % . � > � 9 * % *

9 * U � . � � 9 � � 9 � % � % > � 9 � � % 9

� U � � � . � � . % � > � 9 � % � 9 . * � *

� � > � U � � * � 9 � � % � � � � * � . 9

U � % * > � 9 * � * � . � . * � . * * * > .

� 9 U � * > 9 � 9 � . % * % � > * * * � � . . � % . %

U � > � * > � � > � � * % � � � * * � � *

9 � * � 9 9 � � 9 * > � � � * > � * > � � * 9 .

� 9 % 9 � � 9 9 . � � � % � � % � * � � � � . � � *

� � � . * % > > � � 9 � % � > % � � U � * > � * % U

9 * � * � � . � � * > � % * � � � 9 � �

> � � � � �

� � � � � * * > * � 9 � � * � � . 9 * % >

� . % * � U > � % 9 . % > * � * % % * * �

� � � � 9 � * � � % � � * % 9 * � % � > � � 9 �

% U * 9 U U � % � > � % > � � � � � > *

� > . � � 9 > � 9 * � � � . � � � � � � � . * � %

9 9 � 9 % � � > � � � � . 9 % � * % 9 % � � � . % � > � � .

� � 9 * 9 � � U � � > . % � > * � � . * * � � � * � > . % � �

� � � % � > � 9 . � * > � * � > � � > * � � � * > �

> * U . � U � * > . % . � 9 � � 9 U U � � 9 > . � % � � � � > � � > 9

. 9 � � � � � 9 � * > > � � * % � * U � � * � � * � 9 � > . . % > *

� U . 9 U � > . � � � 9 � % � * � � � 9 U > � � � % 9 . � �

� > > � 9 * � � > � � . . % * � U � * > % � . % � � > � % 9 * > � *

* > � � % � � % 9 . � . * � � � U � � . * *

� > � . 9 9 � . 9 � � � > . * % � * � * � > >

� � U � � � � > % � � % > � * * � * � * � * � * >

* . % % � > % � % � . % � . % > > � * � � � *

� � > � � � � . % � � U � > � � � % � � * . > � . * > � � . %

� � * % 9 > � . * % � � . � > � � * * � *

* U � � � > � � U � * * � � > � > � � * �

% U � � � > � * * � 9 � * > * * > % � % � 9 � > � � * . U �

Page 10: Goldrei.classic Set Theory for Guided Independent Study

� � � � � " � " � " % � � % ,

� � 2 " 2 6 " % 2 " � � 2 %

" � � , � � � � % " , 6 " % 6 " % � " �

% " � � " % � 6 � " % D �

2 2 2 2 � % , " � 2 % � D � 2

% � � � 2 � � , � � � , � 2 " � � 2 � � " � "

� 2 2 � 2 " %

2 � 6 " � � � " " " % " 2 " � ^ D ^

� � � , � 6 " � % , " 6 � 2 " � " D " , % , " D " " D , �

" � 2 � " % " 6 � " , 6 � , " % % ^ % % " % D " � " 6 "

D � � 6 " " 6 " % , � ^ " � � ^ � 2 ^ � � ^ " � D % , �

6 � ^ % % 2 " 6 D , � �

% % , " %

, " % % � � " % � 2

� � � � � , " � � " � � � " , � " 6

� � � � " 2 � � " % % � � % � � � %

� � " 6 � � � % 6 � ^ � �

� D � � 2 " � � 2 � " % � 2 � � " 6

� � " � 2 � � " 6 %

� % " � " " D " % � � � � � � , 6 , 2 � � ^

% % 2 � � , � � 6 , � 2 � , � % �

2 2 2 2 % 6 " % � � , � � 2 D % " � � �

2 " 2 � � � � , � % � � 2 " 2 �

" � 2 � � � 2 � " � � " � " % � �

" " % 6 % � 2 6 2 � " % 6 � D " % 6 %

" % 6 � � " % " � "

� " " " � 6 D 2 � ^ 6 � � D � 2

� � 6 " % � � � " % % 6 � � , 6 " � " 6

� � " 6 " D � ^ D � 2 � � " % 6 " 6

D 2 � � % D �

� 2 6 � 6 ^ D � � � 2 , 2 6 � , 6 6 % �

" � D " � � � , % " � � D D ^ � � �

% � � % � � � 2 � " 6 D D % � %

� � 2 � " % % D �

" %

" D �

2 � " % % � 2 2 " 2 D � � D 2 , 2 " D

� � � � " , 6

Page 11: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � )

� � �

� � �

� � � r � t �

� t � � �

� � � � � � � �

� � �

� �

� � � � � � � � �

� � � � �

� �

� ) � � � � � �

� � � � � � � t � � �

� � � � � � � � �

� � � � � � ) � � � � � � �

) � t � � )

� � � � � � � � � � �

� � � � � � � � �

� � � � � � � � � � � � � � ) � � �

� � � � � � � ) �

� � � � � ) � �

) � � � � � � � � � � �

� � � � � � � � �

� � � � � � t � � r

� r

� � � � � � � � � � � � � � �

� � � � � � �

� � � � � � � � � �

� � � � � � � � �

� � � � � � t � � � � ) � � )

� � � �

� � � � � � � �

� � � �

Page 12: Goldrei.classic Set Theory for Guided Independent Study

� � " # # # � + � # � + # � +

� G # � # � � +

� # � � + #

� # " " � h � # " " � + j �

� � � � � # " " � � � n # #

� � + � � �

� # j # � � � # n �

� + � � # + h G # " �

G j # + " j + h n

� # n + j n n + � j � n h + h

#

+ � � # +

# � + � h # j j + � n G G + h

� � �

# n + � + � n G

� n

� n + � h � # # G � j �

j � � # j h # " + � � # j n + � h + � h n

� # " � # j j

� # # # � j + j � � + # h n � # � # j

� � � # � � h � � � + � G � h # # " � n � h

� + � # � � G j # " " # # G + " #

� " � G � + # + h n " h � h + h � + n + � h �

� # " � � G # n j � # " � h j + � j � j n h G G h

h j � � n " h n + # n # G + G h + # " � � h n j # " � h

� # + n + # j + h n n n + h + # + j � h n

� � # G � + j � j � n G G h j � j � " � n

+ � # h n + G h j + � j n h G G + # + n + � h n # " � h

G " " j n � # n + � G " " � + � n

#

Page 13: Goldrei.classic Set Theory for Guided Independent Study

� � � ( ( ( + , � , � , 3 ( 7 �

, � , , 3 � , D E ( E � ( ,

+ E � , ( R ( R R � , E ( , � ,

� + 3 + (

3 ( 7 � , � ( ( � , � � � � R ( ( ( � ( , � ( E

� , E

(

( , � + ( + , + � E ( � 3 D , ( 7 7 D R + � , �

( , � , 3 E � E ( � 7 � ( � + R ( � E D D + , ( 7

� E R � , (

� D ( , ( D + D + (

+ ( D ( , ( + + (

� ( D + ( + D D + (

+ � E ( D + R D

( + , , ( + R � D +

� + 3 ( + R � + , D

Page 14: Goldrei.classic Set Theory for Guided Independent Study

� � % ( % + - . 2 ( + % ( 4 ( % . 7 2 4 - ( 4 7 % : ( 4

+ . % % : ( - ( 7 % 4 : ( % 4 + % � � % . ( % 4 % - ( - ( 7 4 (

2 - 4 � + - ( � � � - % - 7 ( : 7 % . 4 ( 7 2 + -

� % % � . . . � : % . 7 ( . ( - ( 7 - 7 7 - � % 7 2 7 7 - % 4 7 - 7 4 (

% � � . � . : . % 7 % 4 + 7 - - ( ( . . % 2 - ( 7 2 + - � 7 2

� 7 ( - + 7 � � . 4 ( ( 4 ( - . + 7 - 7 4 +

% . 4 : ( ( - 7 � � + 2 7 4 - � ( 2 4 2 ( ( 2 ( ( + - + ( 7

: + : 7 ( 2 - 4 2 % � + � 7 ( . ( ( � � % % ( ( . ( ( ( + (

4 - . ( : ( + - - + 4 - ( � 4 ( - ( � � 2 ( 7 2 + - . 4 % - ( : (

- : - - � � � + ( + - + 4 + � . ( 4 7 2 7 % + 2 + %

+ . + 7 7 4 � � 7 + - % - 4 4 +

4 - . + ( % - 4 - - 7 7 + +

7 - � % - 4 7 - 4 7 2 4 7 7

7 7 4 : � + ( 7 7 � - 7 - ( 2 7 4 + � (

7 ( ( - 4 � 2 7 7 7 - % 7

+ + . . ( 7 - 4 - - 2 - 4 4 7 7 - 2 �

- � + . 4 7 - + � ( 7 4 ( - - + ( 2 +

: ( . ( ( ( + � - + 7 7 2 � + + : % 7 + ( - + 7 7

+ - - ( + % � 4 : 2 ( % + - ( 7 7 � % 4 : ( % 7 4 ( 4

7 + . ( . ( + - 7 + - ( ( - 4 ( - 7 7 - - ( 7

( : ( 7 � -

- + ( 2 7 7 - � - 7 7 - � � � + + + 2

% � + 4 + �

7 7 ( + - 2 + � � + 7

2 7 � 7 � � 4 7

� � 4 7 7 + 7 4 � + + 7

+ 4 7 � 2 2 + - 7 4 7 + + 7 + -

� + 2 ( + � 7 7 ( + + - - � 7

- 2 ( 7 � - : ( 2 �

( 2 7 7 4 : 2 � 7 2 + - 4 7 7 - ( + 7 - %

4 2 ( � - 2 ( ( + � 2 7 2 2 (

� 7 - ( 4 � . 7 2 7 + + + 7 4 � - +

7 � ( 4 � 7 - ( � 4 7 7 - 4 7

7 7 4 + 7

7 ( 7 + % 2 � � 7 7 + ( + � 7

7 - 4 7 - 7 ( � 4 7 7 -

7 - 2 7 � ( + � � + 2 - ( ( + 2 +

7 4 : ( % � + 2 7 7 �

7 4 : 7 4 4 7 - ( % 2 ( % � - ( - 4 ( 2 . 7 4 7 - � ( 2

: ( � ( 4 ( 4 ( - 2 % 7 - % � - 2 . % - 4 : ( 2 7 ( 4 + ( ( 2 (

Page 15: Goldrei.classic Set Theory for Guided Independent Study

� � � � � " � % � % ( � % ( � , "

� � 0 ( % ( � � 0 % % " � �

0 " � � � � � % " � � = 0 0 , 0 0 0 0 " � = % ( � �

( � � 0 ( 0 " � ( � % � , , � % " � � 0 � 0 � 0 , 0 , " �

� � � � 0 ( � , ( � � 0 0 % ( � % , 0 � � � 0 � � % � � I

0 " � ( I � I 0 I ( � 0 " � = � � % ( , � � �

% ( " 0 ( 0 0 % " � � � , " 0 % = I " , � � " ( , � � , = � "

� % ( ( � 0 = % ( , � � � I � � � % � ( � %

� � � ( 0 0 " � % ( " 0 0 = % " I % I � % % ( % I

0 % ( I % ( , I " % � % = 0 , 0 % I % ( 0

( , = � � I � " % ( % ( , � � , = 0 � 0 � � " " = % " I � � , = 0

0 � " , � I = � % � � = I % ( I 0 � �

� % ( I 0 I � � % � % � % � 0

� 0 � 0 I ( " 0 % % ( � " I I " � 0 " � � I ,

� � � % ( � " � = � � � � 0 � , = � % " I � � �

0

" 0 , � " " I � � " , I � % = , I � = % " � � � � , = %

� I ( , = % " I � , = 0 I � % = , � � " � 0 " = = " I � � � , % " I � �

� " , 0 " ( I % ( , � , % I = % , = " I I � % I I 0

( , , � 0 0 0 I � � % , = � % " I � � � 0 % I % ( � , % I I " 0 % I % I � �

I " % 0 = %

, ( � � " � % " % I 0 � I % � I = � 0 % � � , � , I % � I �

( , 0 " � � = � % " I � � " " 0 � 0 % ( � � � � � % " � � � " � �

I ( � � " = � " � � 0 � = " � = " � � I % I

I 0 , % I , , % , = � � , , � � � I � 0 , % % % , � = ,

, " � % , , " � " � 0 % I , , � = , � � � % I , � � " " I � I

" % I � I � , % , % 0 � % I % � , , = , � , , � %

I " 0 , 0 0 % ( , , = , � , , � I " 0 � % " " I � " 0 � � , �

� " 0 , , % , ( , , � � � % " " I 0 I = "

, � % I � = , , � = � " , = � % " " I � I � � " = = % I � % " � � I % ,

% 0 % � , , " � % ( , � � " � ( I % I , , � � I = , = �

% ( , 0 � � = , % " , % I � I � % ( , 0 , � = % " I � � I " % � ( , I = , �

� % " I , � % " I 0

, 0 � � � % = , � " I � I � , � , I , � � � % � " I � I , , " �

I 0 = % I � � % , , I % % , � I , , 0 � � % 0 %

, % , % % , = � " 0 , I , I � % % "

I , 0 , � % � % I , � " % I � % I % % , , � I

� , 0 ( � � , = , � , � I

Page 16: Goldrei.classic Set Theory for Guided Independent Study

, - . - - 4 9 9 = = 4 4 9 -

- . 4 - - ^ . 9

- = - ^ = , . ^

, , ^ � = 9 . ^ - - 4

9 ^

- . - . ­ 9 ^ - 9 . = = - ^ � . =

-

9 - . , . 9 ^ . 9 - 9 ^ - ^

, - = . , = = ^

, = - , 9 ^ . = . 9 Ì ^ , . = , - - ^ , ^ ^ .

. , - 9 . . = , , 9 - - -

. ^ ^ . ^ ­ - . , . = . 9 . = = , - , . =

^ . 9 . � Ì = . , 9 ^ . 9 4 � . 9 ­ . = �

4 = = 4 ­ . ­ 4 Ì 9 9 4 . , , 4 ­ Ì 4 Ì 9

, - - , 9 - . , 9 9 ^ Ì . 9 ^ ^ . 9

- , 4 ^ Ì . = - , - 9 9 9 � = 9 � . 9 , 4 . =

- 9 . = ­ Ì , - ­ 4 . = � - � = 9 9 - 9

. 9 4 = ^ . 9 4 - ^ , . = ­ Ì . 9 4 4 ­

4 ^ . = -

^ � ^ . = Ì Ì - 4 , 9 4 , -

. ^ . 9 �

- . = - �

- . . =

. � Ì = , . - - 4 , ­ 4 � 9 � - 4 ^ . 9

. 9 . Ì � - - � 4 - 4 9

4 � - Ì � Ì , 4 9 � � ­ 9 , 4

4 Ì - 4 - � 9 9 . ­ 9 9 � - 9 4

9 . 4 Ì - � � . ­ - 9 . 4 9 - �

. . - Ì � 4 � Ì 9 , � � 9 4

- ^ 4 ^ 4 ­ � � - 4 9 9 � ­

. 9 - . Ì � ­ - . = - ­ . ­

Page 17: Goldrei.classic Set Theory for Guided Independent Study

� � � � $ � & � $ * .

� 5 � � 5 � $ 5 7 . $ 7 5 . $ < < 5 � < � $ 5 � * . * 5 <

$ � � < � . 5 � � � < < $ � � � & . *

. � 5 � � < 5 5 � . * 7 $ & 5 � � � * � � & . *

< � < � 7 . * � . � � . � < $ � � < � � 5

� � . < � 5 5 7 � . � � 5 � 7 $ $ $ . 5 $ 7 * & . . * . � �

. � � � . . � � � 5 7 $ & 7 * . * 5 < 7 �

& $ � � . $

. 5 � . � . 5 $ 5

� 7 5 & . 7 . � $ � $ 7 < . < $ $ 7 .

5 � 5 7 * � � �

7 5 � & . 7 � . . 5 7 7 * � . .

7 . 5 � 5 � * . & 7 . 5 $ . 5 $ � � < 7 & � . *

� 5 7 � $ $ 5 � � . * . � . & 7

� $ � * * $ $ . 5 7 � . < $ $ 7 . � � 5 � � 5

< $ � 5 � � . * 7 & 7 5 � < $ . * & 7 5 5 . $ . . * . 7 � .

7 . 5 � 5 5 � . & � 7 $ . 5 7 � . � . 5 * 7 . *

< . 7 . 7 � . � 7 5 7 &

. 7 & . $ * < 7 & < & 5 . * $ * 5 < 7 & � & * $ & 7 * 5

7 . . . < & � 5 5 7 * * . $ 7 & < & 5 � � < * � & *

5 7 & 7 . 5 7 * 5 . 5 < $ 7 .

� � * . & 7 $ 7 5 . * � & 7 < 7 * . * 7

* � 5 & � $ � . $ $ 5 & < � 7 . & 5 7 &

7 * 5 < 7 � � $ . 5 7 < 5 & 5 < < 7

5 $ * * 5 < � 7 7 * * 5

. . 5 � 5 < 5 < & 5 7 5 7 � . � 5 < .

5 7 . $ & 5 & � & 7 � . & . . 5 < & �

� � � . & 5 < . * � 5 < 7 & � � $ $ & . * & 5 < 7 & <

7 * . �

. * . * $ . & � 5 7 & < & 7 * 7 � 7 & 7 $ 7 * * $ 5 7 < 7 $

5 � � 5 $ � . . < & � . 5 5 < 7 5 7 5 7 � 5 . * 7 . $ 7 $ 7 &

5 � <

� 7 5 5 7 $ & � 5 < . < < & � . 5 5

Page 18: Goldrei.classic Set Theory for Guided Independent Study

� � ( � ( 2 ( � ( < @ B � �

( � < F ( � F < � � ( � B �

� ( � � � < F � � ( � ` 2 F ` @ ( @

� � � B @ B � ( B F B � ( � < ( � ( k �

F � � F � ( 2 ` ( 2 � � F ( � � 2 � F

F ( � � F � 2 B � 2 ( B < � � ` ` ( � B @ ( � B ( k

( ` F ( ` � < � � ( � ` B @ `

2 k k ( � � @ ` @ � � � ( � �

@ ` � F � ( � 2 k � 2 � F ` ` � � � ( ` � k @ ` @ �

@ ( @ < � F � ( � @ ( � ` @ B � B � F

� ` � � F 2 ` k k < � k k k �

� < � F ( � B � ( ` F ( ( ` ( 2 < @ � � < ` B � `

F ` @ B 2 ` ` ( k ( 2 ( k � � F ` � @ �

2 2 � F < ` 2 � < ` � � ( � � F @ ` � < k ` � < `

B

� < � F ( �

B � k � � B 2 ( k B 2 < k B ` k ` ` 2 � � < � ` ` 2 <

F B F � k ` � � <

F ( ` ` ( < B B 2 � F ( � � F ` @ ` � < `

2 ( � @ ( k @ � F � ( � � � F ( < B

( �

� < � � � F ` ` � ( < � � B <

( ` � F ( � � ` @ ( � < 2 ( k � � � ` < � 2

Page 19: Goldrei.classic Set Theory for Guided Independent Study

� � � � � " # � � ( � # � � � ( �

- � � � 3 � � � � # � # � 3 � � �

( # - � ( � ( � � ( � 3 � � � # � � #

# � � ( � ( � C � � ( # � # � - � � � 3 � # # "

� 3 # # � ( # � � 3 ( � � � � ( C ( " � 3 "

( � � ( � ( � � � " C 3 � � � # � C 3 � # �

� � # - � # # � � 3 � # � � � C # # � 3 � 3 � �

� � � � � � 3 3 � � 3 3 3 # � ( � # � # �

� ( � � � � ( � � # " � # 3 � #

� � # � � � � " � # # � ( # � 3 �

� � � � � 3 � � � ( # � 3 �

# � � ( � � � ( � � C # � 3 C # � - � �

( # � 3 - � - � � � � � C � 3 # � - C � � # � 3

� � # � � � ( � ( � � " � # � � C � � ( � 3 3 � C � �

# C � � � � � � # � 3 # � � C 3 3 3 # # � � � � � �

� # - � - � # � 3 " # # " � �

( � � 3 # � � � 3 � 3 � � � � - � � � C # � � �

( " # � 3 � 3 � � � ( C " # � � � ( � " � � � C � 3 �

� ( � � � ( #

" � � " 3 " � ( C � � � 3 # � � �

( � # � 3 � � � # 3 3

� C

3 " � � ( � � � 3 � � # � " - � � 3 3

(

� 3 " � � � ( � � � � � C

� � ( ( � � ( � � � � -

( � ( # � � � � ( � 3 ( � #

( � " � 3 � " � � ( � � 3 3 � # � # � �

# � 3 3

� 3 3 � � C

# 3 - C � ( "

� # � 3 #

# � ( #

Page 20: Goldrei.classic Set Theory for Guided Independent Study

� � ! , 4 < � �

C ! , ! H < 4 , 4 4 C ! C , C N ! 4 !

4 ! C � C ! � , C ! < , !

< C , < C , N 4 ! C C

� � s � � , � , C ! � � , 4 �

C < H � C < 4 ! < ! < , !

C ! C , � H ! C ! < ,

< C < , � , , 4 < < C , N 4 ,

< H < � < 4 C H ¬ � , 4 , , < , !

! ! < C 4 ! , C , �

! < C � , ! , 4 C C < , ! � , 4 4

C 4 C 4 ! < C < , , 4 4 s

� ! < , ! < ! < � , 4 H 4 C C � � ! C ! 4

C ! C

, ! C C , 4 ! � , 4 � C <

s C , H � , 4

< C C < � C C < < s C �

C � ! 4

� ,

� 4 4 � C

< ! < ! < C ! C , � , 4 , � � � , 4 , , 4 C

� C � C ! H ! � , 4 C ! H ,

! < C 4 , 4 C < 4 ! ! , C ! < , C !

! < , ! < , C , 4 , ! �

H < < ! C C H , H

< , 4 s s C ! C

� C < ! ! � � ! ! < , ! H N , < C , 4 !

< , ! � C < < ¬ � , � � 4 � C ! C

< � ! , !

< C � � C ¬ < N C ! � C ! <

� H H 4 N , C < C C ! < C , 4 4 C ! C ! ! ¬

< , C ! C C , � � < ! � C 4 , !

C H C ! < 4 ! ¬

Page 21: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � ! � � � � �

� � L ! � � � P L � � ! � ! � ! � P � P L P � � � � L

� L � ! � L ! P L P � � � � ! j L ! � � L � ! �

� � � � ! � L P � L � ! P � � � ! � � � j � � !

P � � � � � L ! � � � � � � � L ! � j L P ! �

� � � � ! � � � P � ! � L � L

� � L � � L � ! j � � �

� � !

� � � � ! � j � � � � � � � � ! �

� � P � � � ! � ! � j � � L � � L � � �

� � � � � � � � � �

� � � � L � � � � � P L � � � � P � L � !

� P L � L � � j ! � ! � P � � � � j L �

� � ! � � ! � P � P L � � � P � L

! P ! ! � �

� � � P � � � j �

� L � ! � L � ! � j � � � � � j � P L ! � P � � � ! �

! � � � � L � � ! � � � � � � ! P � P � � �

� ! � j ! � j � � � � � L � ! � � � � L � j P ! �

L � P � � ! � j

� � L P � � j � � j � ! j L P L P ! j � L P

! ! � ! � � � � � ! � � � ! � � L � � L � � � � � ! P � L L P P

� ! � � ! � � � L � L L � � j � ! P L � � � ! � j L P L

� � � j L ! � � � � � P � j � � � � � �

� � ! � �

! P

P

� � � � � � � � � � � L L � � P

� L �

� � ! � ! ! � � L P

� ! � � � � j � � � � � ! � � j ! � P ! � � � � ! � P ! �

! P � � � ! P � L ! � � � � � � ! � !

� � �

� ! P � P � � � � P � L P � � � � � L

Page 22: Goldrei.classic Set Theory for Guided Independent Study

� � � # � ( � . 0 � ( � � � �

0 . 0 � � . � . � � . C .

0 � . 0

� � . . � C � 0 ( 0 ( � .

� ( 0 . � �

( . ( � ( 0 ( �

� 0 ( 0 .

(

� � � ( C 0 ( 0 . . 0

(

C

( C

C � º � ( ( � ¼ � C º # � � ( � # (

. 0 . 0 0 � � º � � ¼ ( . 0 ( � C 0 ( 0 � 0

� ( 0 � � ( . ( � . ¼ 0 ( ( C 0

¼ � # . � � � º # � ( � ( º ( . . 0 ¼ �

� 0 0 � ( � º � º � ( � 0 ( � ¼ C 0 ( � ( � ( .

( . ( � � ( 0 ( � ( ( . ( . ¼ 0 0 � ( #

C � ¼ # C º º � # ( #

¼ � ( � 0 � C 0 � . # � 0 ( C # � � �

( � ( � � C ¼ 0 C 0 0 � # ( � ¼

� ( ( ( � � ( . � 0

º � � #

º � � C ¼ � � #

º � C 0 # ¼ ( � ¼

� ( �

� � 0 ¼ � º # 0 ( ¼ 0 º � ¼ ¼ ¼ . ¼ 0 �

( � C 0 � º � � C ¼ C � ¼ . . º � ¼ º � C ( � ¼ 0 ( . . ( 0

� 0 º � . 0 ( � � ¼ ( . ( ( . 0 �

Page 23: Goldrei.classic Set Theory for Guided Independent Study

� ! & ( & ( ( , & ( / ( , �

/ , ( / & ! , K ( K ( ! (

& K / � & ! � / & K

, � ( /

& / !

& ! / , � �

/ , �

& ª &

& / ! ! �

/ ! / , �

& / !

& !

& � , � K ( / , � � / ,

& / ! !

& ! !

& /

ª & ! ! , � & ! ß , ( � , � / ,

& / ! !

ª &

, ( ( , � ( / � / / ! / ,

& �

� ( , ! ! ( � / ( , ( & & ( � , �

/ , ( / & , � ! / , /

, � ! , ( / /

! & / ( ( & & , ( ª K ß & / /

& ª ß K ( , � ª ! ( ( & & , / ! ß / & ! K & � ! ! ,

& ß � / , � / ( , � & / K K & & , / � / & &

ß ! ! � & ( K � ! & / & & ( ß & ( (

� ( , / ! ( � , � / � ß , / , ( � & , � &

/ & , ! ! , � ( / ! & ( ! , ( / , � & / ! ( � , � (

( , / ! / ( K � ( � ( , � / �

� & ! ª � ( & & K ( , � & ( / / !

ª � / � / � / , / & / ! & � / ( / ( / ( / K / ! (

/ ( , ! / , & � ! ( � / , � / K /

� , & ß ! & , ( & ! / , ª , / , / , ª ( , � / ß

, ß & / ! & ( , � / � & ß / ! ( � , ß & ( ª / , / ! &

! , , � ( & , ß & ( , � / , � , ( / , ß ! ( /

& ! / , ! & ª ! / ( , � , / ( ß / ( ª / ! ß , / ( / , K (

& , ª � � / ( / ( / , ( / & ß / ! ß & � ! , ,

/ / & ! & ( ( / , / , K � / ! ( ( , �

� , / , / , ( / / , ª ª , & & ( ª / ª ª

Page 24: Goldrei.classic Set Theory for Guided Independent Study

� $ % & $ - % % & 3 3 - 7 % % &

& $ % � % - 3 & & & % M

% � & % & $ M � $ & 7 3 $ M $ M % % 3 $ % $

% � % � % - 3 $ % - � $ M % t $ $ % & $ $ M

� $ M � 3 % � $ M $ - 7 & $ % t �

& � & $ M M � $ M % 3 $ % t � % 3 t � % &

$ & M � % - % & & $ % 3 � $ M M M $ % % � % 3 % % &

$ % � 3 - � 7 & � 3 % & � � � t % �

& & $ $ 3 % & 7 3 � - % & -

& & 3 $

& & 3 t � $ � & % & $ % � � 3

$ % � % $ � % & - 3 t

� & $ t �

% & 3 $ % & � % 3 % t t & M M � & t

& & $ $ $ 3 t & $ $ M % 3 $ t � % 3 % -

� $ % � % & � % � & % $ % & & � & M � � � � % & � � % - �

$ % 3 � 3 % M � - 3 $ $ M � - & $

% & - $ & � $ M $ $ M � % � $ 3

& & & % - % & $ % - M - 3 & $ % 7 � � $ M % % � 7

7 � 3 3 $ 3 M M - � 7 % - $ & % 3 % $ M M $ 3

� $ 3 � t � $ & � % & � � % % - $ M M

$ % & � � � 3 - � % 3 % � M 3 3 �

$ � 3 % - � � % & 3 � � % $ $ & - 3 $ M � t �

& & & 3 � % & % & & $ M t % � $ $ M $

$ M M

% - � & - % $ � � % 3 � - % & � � 7 � % 3 - 3 � � M

M $ & & 3 3 $ M M

Page 25: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � �

; � � � � ? � � � B � � D ; � � � � � � � �

� � � � ; ; � � � � � D ; � � � � D � �

� � � � � � � � � � � � � B ? ? � � � ; � ; ? � D

� � ; � � � � D ? � � � ; � � � � � � D � ; D

� � ; ; � � � � D D � � ; � ? � � � � � � B �

� � � � ? B � ; � � ; � B � � D � � B D � � D B � D � B � D � � � D D � D

� � ? � � � ; ; �

� � ? D �

B

� � D ? D

B

�� ?

� D � � � D ; � D B � ? � � � � � � � �

; � � � � D D � B B � � ? D � � � D � � D

? � B � ; � � D ? � � D � D D � D � D D � B ; �

� ; � � � D � � � D ? � � ; � D ? � � � � � � D � � � � � D �

� ; ; � � � D ? � ; � � � � ; ; � D � � � D � D � � � � ; � � D �

� ; ; � ; � � � � � � � D � � � � � � � D � � D � D ? B

� � � B � B � � � � � D � � � � ? D � ; � � � � ? �

D � � � D D � B � ; ; � � ? D � ; � ? � � D � � � D � � D

D � ; � � � D � � � ; B � B � � � � �

� � � � � B D � D � D � � D � � � � B � D � � ? D D ?

� D D � � � � D ; � � ? � � � � � � D D � D � ; � � � � �

� � ; � ? � B � � ; � � � � D D � � � ; ; � D D D � � � � � � ;

� � D � B ; ; � D � � � � D � D � � � � D D � D D � � � B D

� D ? D ? B � B � � � ? � B � � � � � � � � D D � D

� D � D � � � B � � ; � � � ? D �

� ? � � D ? D B � � � � � �

� ? � ; ;

� � � D � D � � � D � D � � � ? ; ; � � D � � � � D ? � B

? D � ? ; D ? D � ; D B � ; � � D ? D B � � D ; D ; B � � � � � � � D

B ; ; � � ? D � � � � ; B B � B � � D � � � D � D � ; ; � D

� D � � B ; B � � B � � � � B � � D D � D � � ? � �

� D � � � ; ; � � ? � D B D � � � � � � � B ? � ? B ; ; �

Page 26: Goldrei.classic Set Theory for Guided Independent Study

� � � ! ! � " $ ( " * $ / !

! $ $ � ( G $ I * $ / " � $ / / $ ! � $ !

� G ! G � ! � G $ *

� � !

G

� $ ! G I / ! ! $ ! $ " / $ !

� ! � $ $ ! " ! ( $ / $ � " (

$ * $ � ! � / G G

$ G $ �

/ $ � I ! �

G $ / $ � $ I

$ ! �

G � $ ! ! � $ ! � ! $ " "

� G / " ! $ I � ! G $ G $

$ / " $ G ! G G $ I � " G G " G

G $ / /

� $ $ � ! $ "

� � ! G $ ! � " G I $ � ! � $ ! G $ / /

! $ G $ ! I ( $ / ! � G " I (

� � ! ( / /

! � G � ! � $ G $ /

� G $ / /

! G $ / $ / ! (

$ $

G ! � ! ! G $ ! I ! (

/ $ ! * $ " ! G ! I ! � ! ! " $

$ ( $ $ / G ( " I $ / " / $ G I

/

Page 27: Goldrei.classic Set Theory for Guided Independent Study

' ( ) 7 8 9 7 ) 8

) 9

9 7 ) ) 9 ( 9 � 8 ( ) (

9 ) ( 8 ( 8 ) 8 8 ) � 8 7 ) (

) 8 8 » ¼ 8 7 À ) À 7

(

9 8 9 ( 8 ) 8 ( » 7 � 9 9

( ) ) 9 (

( 9 ) 7 8 ) 9 À 8 ) 8 (

8 ( ( 8 8 7 9 8 » 7 ) ( 8 ( 9 » 7 À ) '

) 8 ( ) À � À ( ) À 8 9 7 ) � ( ' 8 »

» 8 8 8 » 7 ¼ 8 8 » ¼ » 9 » ( 8 » » À

) ( 8 8 9 � ( À ) � � 8 9 ' ( 9 ) ) ) À ( 7 7 À ¼ 8

9 ( 8 ) � 9 À À ( ) 8 ' ) � ( 9 9 8 » ' 8

8 7 � ( ( ( 7 ' ¼ 8 »

' 8 9 � ( 9 ) ) ) À

8 9 ) 8 8 9 ( ( 9 9 8 � � » 9

) À � ) � ( ( � ) ( À 9

8 9 ) 8 9 8 9 ) ) ) 8 7 8 ) ¼

) � ( 8 8 8

8 9 7 ' 8 9 )

( 9 8 9 8 » 7 � �

9 ( 9 » 8 8 ' À 8 9 8 9 ' ( 9 ¼ ' 8 ' ( ( »

» 8 9 7 ' � À À ( ) 8 ) » » 8

) � ¼ À ( 8 ) 8 9 ( À À (

» 8 9 � À ( ' � ( 9 8 ) 8 8 9 ( 7 8 �

( � 7 8 9 ) 8 8 � 8 À ( ' » 9 ) À À 8 '

9 8 ( 8 9 8 9 � ( ( ( 8 9 8 9 À 8 9 9 8 8

8 8 ( ( 9 9 8 » 9 8 9 8 ( (

À 9 7 8 ( ( 9 » � � 7 ( 8 ( » (

8 ) 9 À ) 9 ( ( '

� ) 7 9 ( ( » 7 8 ) ) ) 8 9 À ) )

' 8 9 ) 9 8 ) 8 » � �

9 9 ) ) ( 8 9 ) 8

Page 28: Goldrei.classic Set Theory for Guided Independent Study

� & ) � + � & ) & ) �

= � ) � + & ) � ) � + + � +

& J ) � � + + J ) + � & J ) + ) + ) � T

T & ) ) J = � & + ) = =

) � ) ) � J ) & = = ) + ) ) � +

) � + T + = = J ) � & ) � + ) � + & + ) T = = & ) = & +

+ + ) � � +

� T + + � � & J & =

& J & = =

) � J T + + ) & J

) & & = =

) + ) ) J = ) & & = T ) + =

+ ) � = T

� � J ) � ) � & ) & = =

& ) + ) � ) � J & = = �

T J

� + ) � + + = ) & J = & ) Ã ) � T

+ + + + ) & = J + & ) � ) � ) ) Ç +

T Ç + J � & & & ) + ) & ) � J

� = ) � Ï + & Ã = + + � + + T ) + )

+ ) ) � + &

Page 29: Goldrei.classic Set Theory for Guided Independent Study

� ! � $ & ' ! .

� ; � ; !

' ; ; ! � L $ ! &

' � ;

; ; ! ' & ! & $ x � ; & & { L $ � { '

! ! & � ' � � ; ! !

; ; � ! '

! & !

;

! $

& ; � ! ' L $ ; ' ! $ ' { & ' � ! x � '

& ' � � { $ ! & ! & ' � ; ! ! & & � ; x $

L & ! $ ! � ! & $ ' x ; & { ; � ; x $ ! x � { '

; � ! ' $ ' � ! ; ' . ' L ; ! x x

; � ' ! � { ! � { ! � $ ! ' ! $ & ! x &

' � ; � $ ; ! ' & ' $ & ! ! ' ' � ; ! � { L

& ! � $ � x ' ! ! $ � ; ! �

! &

� ;

& { ! &

� ; � � ; ! ' � ; x $ ' ! $ ! � !

' x ' ! � ; & � ' ! � $ '

' ' $ & x ! � ; $ $ ' {

& ' ' ! ; ' x ' x

$ & ' ' $ � ; ! � � ; x � ! ' $ ' � ! $

! ; ' ' ; ! $ x ! $ & x

$ � . { � ; ! & & ; x { ! & ! ' '

$ ' � ! � ; ' $ $ � x ! & & x � ; ' !

$ ' ; { ; $ ' ! � ' x x ' ! $ ; $

� ; x & � $ ; x ' $ ' & � ' $ . � x ' ' ! ; $ $ L ' ! x

Page 30: Goldrei.classic Set Theory for Guided Independent Study

( * + * 1 2 1 8 1 ( 8 2 8 B 1 B 2 1 G ( + B * 1 *

1 1 G B + ( 8 2 8 (

( * + * 1 8 \ 8 G G * ( 8 8 8 ( 1 8 +

( 8 ( 1 * 1 G ( + B * + 1 *

1

* 1 8 1 2 * B \ 8 1 G ( 8 2 8 � ( 1 2 B \

( G G

� ( B G * 1 * B \ � B * + 8 \ 1 G � G * + 8 B

8 1 * � B 2 1 G \ G 1 G 1 * B \ \ 8 1 G ( G B B 8 \ 8 \ 8 2 * \ B 8 8 � 2

G 8 ( * ( G * + � ( \ ( � 8 2 * + 1 * 8 1 \ 8 * \ 2 * B * \ 8

8 1 G ( + G * 8 8 * ( 1 � 2 G 8 2 * B 2 B * + 8 \ 8 1 G (

B \ 2 * G * ( 8 * ( B \ 8 1 G 2 8 \ ( 8 1 * G * \ B * 8 ( � 2

* 8 ( 8 2 ( + 8 � ( ( G 2 8 ( 2 8 8 1 G G � B B

* � � ( ( 1 G G ( G � � 2 * B ( 1 8 1 B G G B

� ( * G \ 1 G G * 1 + 8 * B 1 G G 8 8 � 2 G * 8 ( 1 ( 8 ( 1 G G 8 � B B

1 G * 8 \ \ 8 8 G G * 8 2 B 8 � 2 * + 8 \ 8 � B ( B 2

1 * B \ \ 1 G 1 G ( B * � ( 8 8 � B 8 ( * 1 * 8 1 ( * + 8 2 * * B

+ 8 * B G G 2 * B 1 G ( ( G G B * 8 \ B \ * � * 8 �

8 B \ 1 B B \ B G 8 8 B \ \ 8 8 G B B 2 (

1 2 \ 8 2 * \ 8 * 1 8 G 1 � B 2 ( 2

1 G 1 2 1 2 * + 8 ( G G 8 * 2

+ 8 8 G ( 8 G B B B \ * B B 8 8 B \ 8

\ + B \ 1 \ 8 � G 8 * ( 8 ( 1 8 2 � * B 2 ( B 1 * � B 2

1 G * � G � 1 * B 2 1 \ 8 ( B + ( G 8 \ B 2 B \ 2 1 2 8 * 8 \ B 8 \ *

B \ 1 G G

B \ 2 8 ( ( 8 � G + * B 8 \ 1 B \ 2 8 � G 8 * ( 8 ( 8 \

� B 1 G + � 8 G 2 * B 1 G ( B 8 G B 2 ( * 1 2 8

8 ( * + 8 8 ( + 8 \ * + 8 B � 1 � B B 1 G G B 2 B * + 8 8 2 * 2 * � B (

B + \ B * * B G 8 * 8 8 8 2 � 2 B 2 8 + 8 * 8 \ 8 B � * �

Page 31: Goldrei.classic Set Theory for Guided Independent Study

� � � � � ! # � � # � , � � � #

5 � � � � , , #

> � � # � ! @ � , , ! # D D # � @ @ , # @

# 5 !

!

# � � � �

@ , # # � � � @ � # � D � 5 # � ! @ #

# � ! # � ! D � ! @ @ , @ @ � 5 ! � @ , D � #

@ # � # @ � 5 � ! D # 5 @ � ! 5 !

! � ! # � � � # � � ! � 5 D � � ! � ! � @

# � @ � � ! # # � ! � , ! # @ @ � ! � 5 @

� � � � � � ! # @ � � � � # D � # ! @

� � , @ � � � 5 # � � > ! 5 ! @ � !

� � # ! # ! # # D @ ! @ � �

5 # 5 , ! � # � ! # ! � � # � ! @ �

! � # @ @ , � 5 ! # � @ , # @ � ! @ # @ #

� ! � � , @ � ! � @ , # , , #

� # � � � @ � � @ � � @ @ @ � D , @ � �

# , , , , , � # @ > # D �

# � @ � , 5 ! 5 � @ � # @

@ @ @ , #

! � # @ � 5 � @ , ,

# ! � ! @ �

# @ 5 � 5 @

� ,

� , , 5

� 5 # @ # � � � @ � � 5 # � � # @ � D , � > # � � � �

� # # � ! � ! � @ > � , @ @ > # 5 , � # # @

# @ 5 � , @ � � � 5 @ � # ! # � � ,

# ! � ! � # @ # � � ! � ! # 5 #

Page 32: Goldrei.classic Set Theory for Guided Independent Study

� � # & � # ( * + & . . & # 3 # ( + 5 # � . 3 5 + #

* ( 3 + # . # # * + ( *

� # � ( 3 � # # 5 * . # + # ( & * * 3 # . & 3 # . & + (

& ( 5 3 ( 3 + # � # * # � 5 # * + # 3 . . 3 5 # 3 #

3 ( + & + 5 ( + # 5 3 ( 3 & 3 . # # 3 � # � # # . #

� 3 + 5 ( 5 * 3 � # # 5 3 3

# � 3 3 # . 3 3 3 3 + 3

3 . . 3 ( * . 3 � # . & # + � � 3

� # # + ( # 3 3 & + 3 * * 5 + # . + 5 . 5 & + #

# � # � # � 3 3 ( ( ( ( # + & * # ( 3 * + 3 #

5 ( � � 5 3 � ( # + 3 � + 3 5 5 + ( . + � # 3 + & � � 5 & #

3 . & . + ( ( * 3 3 . # + 3 � # # . & * ( + & +

+ � # 5 5 3 & * 3 � ( 3 + 3 & + � 5 * � # � 5 3 * & . (

5 + 3 � � # ( 3 * 5 * # . # * + 3 # * 3 � & * #

& ( # + � # # * # + 3 # 3 * 5 3

# � # * + 3 & * # 3 * 3 # * #

& 3 5 * ( 3 & . + � 3 * � 3 � + 3 . #

� 3 � + # + 3 # + . # * 5 # # 5

�( 3 3

( 3 . 5 * + 3 + # 3 * ( + � 5 * + # � # & . 3 + # � + 5 3 * .

# � & + & . . # . 3 ( # ( + 3 + # & . & 5 � # 3 + # � # & . 5 + � � 5 �

& � 5 * + � 5 3 3 * & # . # 5 & . # & * ( 5 * ( # & 5 #

& ( + + 3 � 5 + * � � # 5 � # 5 & . # & * 5 ( #

# * ( 3 3 5 3 � + 5 ( . . & * 5 * * 5 + # # � 5 # 3 + # 3 �

� ( * * + # & * 3 # 5 * + # * +

# # * 5 5 3 * 3 & * * * + # ( # � 5 # & ( + & + + ( ( + � # . 5 +

3 + � ( # # 3 5 + ( & � + & . ( ( � # � #

+ 5 ( + & 5 � + � # 3 � # & . & * & . ( + 3 ( 3 + & # #

( & & � ( # # * # 3 � & 5 * . � � 3 * * # + ( # & . # & * 3 * ( +

+ � & . ( # & � 5 . # * . & ( ( 5 * + 3 � # * + 3 & 5 * ( � &

#

+ 5 3 ( 5 . + 3 + � � & . * � ( 3 ( # ( 3 * ( & . + 3 � + #

5 + ( & & � 5 � . . # ( & * + # # * & � ( + 3 � 3 # � & #

Page 33: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � & ( � � � � � � � � � � � ( � � �

� � ( � � � � � ( � � ( � & � �

� � � � � � & � � � � � & � � � �

� ( � � � � � � & � � & � � � & � � & � & �

� ( & � ( � � �

� � � � ( & � � � � � & � ( & � ( �

� � � � & � � ( � � � � � � � � ( � ( &

� � � � � � ( � & � � & � � � � �

� � � � � � � � �

� ( & � � � � � � � � �

� ( � � � � � � � � � ( � � � � (

&

� � � � & � � � � � � � �

� � � �

� � � � � & ( � � � & � � & � � � � � � � � � � �

� � � � � � � ( �

� & � & � & ( � � � � & � � � � � �

� & � � � & � � � � � � & & & � � �

� � � � ( � � ( � � � � � � � & � � & &

( � � � � � � & ( � � � � � � � � �

& & � � � � ( � � � � ( & � � ( � ( �

� � & � � (

� � � � � � � � � � � ( � & � � � � � �

� � � � ( ( � � � � � � � � � �

� � � � � � � ( � � � � � � ( � � � � � � � � � &

� � � � � � ( ( � � � � � � �

� � � � � & � � & � � � � � � & & � �

� � � � � � � & � � � � � & � � � � ( � & � � & �

& ( � � � � & � � � � � � � � & � � � �

� � ( � ( � � � � � � � ( � � � � � � � � � � � & �

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� ( � � � � � � � � ( � � � � ( � & � � & � � � & � (

Page 34: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � ! � # � � � � ( ) � � � � ) / � � � � � / � � �

� ( � ( � � � ( � ! � � � / � � # ) � � � � � ) � � ( � � � ! � ) ( / � � ! �

� ) � / � � � ( ( ) � � � ( � � � � � � ) � � � ) � � � / � ) / � � � � � �

� ) / ( � � ! � � / � ( � # � # � / � # � # � � ! � � ( ) � � � � � � / /

� � ! � � ) � ! � # ( � ( � ( � ) � ( # # � � / � � ) � � (

# ) � � � � ! � � � ) � ) � ! � ! � # � � � � � � # � ( � )

� # � � � � � ( ) ) � / / � ! � ( ) � # � � ( ) � � � ! � � � ) ( / /

� � ! # � � ( � / � ( � � � � ) � # � ( ) � / � � ) � � ( ( � � � )

� / # � � � ! � � � ( � ) � � � ( )

� ) ( � � � � � � ( ) � � � � # � � � � � � � ( # � ( ) / �

� ) ( # �

( � �

( � ! # �

� � � � ! � � � � # � / / � ! � )

! ( � ! � �

! ( � �

( � � � ) ) ( # ) ( # / / � � � � ) # � � �

� � � ) # � ( ) / � �

� � � � ) � ( ) �

/ � ) ( � ) ( � � � � # � � ( # � # ( � ) � ( # � � /

� ! # � ! � � ( � / /

( � ! � � � � � � � � / � � ) ( ) ( � � � ( # � (

� ) � � ( # � # ( ) ) � ( # � � / � / ( � ) � ) � )

� � ( � � ( ( � � ( � � � ( ) � ( � ) ) � � � � # � (

Page 35: Goldrei.classic Set Theory for Guided Independent Study

! % ! , . ! / , 0 % ! , % < < ,

. % . / . 0 . ! / , , 0 / % . % , % ,

! ! % . ! , . , / . ! , . , . ! % . P , 0 0

% / ! , ! < % % 0 . ! , , % . ! / , 0 %

! , 0 < . / % % ! / < , 0 . ! % .

< . ! ! % . ! . ! , . . ! . / %

, % / . ! % . ! % . , , 0 / %

! P % , . / , . , � , / , ! � % % % P % , % / / , 0

, , . % . . � ! % � ! , . , / . ! / , % 0

. ! % ! � % , 0 . / / 0 , % !

0 / % . % % % P P % / . / , . , 0 % �

. , ! . , 0 0 / ! / 0 . % %

/ . / . ! ! � , . % . ! !

. % , 0 0 . ! % / / < . , < , % % . / , . , ,

0 % / , . 0 . 0 < , / % % ! < ! % / % % , %

! / , � / . < . ! , . ! , < % . , , . < , < % , /

! < % . / 0 . ! . ! � . , 0 % P � % %

, % . , . ! % % , / . ! P , .

� % . . ! / % ! , 0 0 / % % . < < , % . / . � / . ! / . ! ,

0 < �

! / / , % � , . / , . < , 0 % , / , % . . / . / , . < � %

. � / ! / , , P � . / % . ! % . ! % ,

. ! / , . < 0 % . / % ! . / ! / < , / , . < , 0 / , %

< , / / , < / . < . ! % . < / / % � < � .

. ! , . ! < % 0 , . ! / 0 / % . , / , ,

/ < . . / < / % < . ! . % . , , %

! , < / 0 / % . , ! . ! / / , / %

! < % . ! % , . / 0 . ! , . P . / ! � / / .

/ , 0 / , , ! % / , . , 0 , P / % 0 . . ! % ,

, , , , 0 / 0 , P . , , � % , !

, / . ! , ! % ! / % , / , / 0

. / � , 0 � . ! / / < % , . ! . �

0 , < . / % , . % � , . � % . ! , .

< % % 0 . 0 < . ! . / % ! % P . . � P .

! % , . . ! / 0 . , , % � % P / . , / , %

! . / % 0 . , . ! ! % , % ! , 0 0 0 .

. ! < % , % . , . . % ! � / / , / % � . / % , / < 0 .

Page 36: Goldrei.classic Set Theory for Guided Independent Study

) * + + 3 4 4 5 ) * 6 +

+ v * + z 3 3 4 6 5 4 v � * + 6 z * 4 v 5 5 5 z

* z ) * + * z 5 z * 5 3 * 5 z *

* * + 4 * + 5 5 3 ) 5 6 5 z

5 + 4 5 4 3 * v * + z *

z + ) 4 6 6 4 z 3 � ) z * 6 z * 3 4 4

* + z 4 �

5 ) * 3 * + 5 ) 3 4 ) 5 z * * + z

6 z * z 5 ) * v 5 5 4 3 * v � 3 � ) z 3 5 ) * + 6 * 5 3 * v z

) * z * + z 3 4 6 4 3 z z z 3 * + v 5 5 3 � 3 5 ) + 6 * ) v 5 3 * ) v z

z + 3 4 4 * 3 ) z 5 z 4 * z 3 * * + z ) 5 z * + ) +

4 + 4 z 5 4 * 6 * 5 z � 4 ) 3 * * + * 6 5 * v � 6 z

4 ) 5 5 3 5 z * 5 z 6 3

v

z 5 3 * 3 4 3 5 6 5 z * 3 4 4 z z z 4 4 + 3

3 * + 3 3 v z * 5 * ) 4 ) + 4 z * + 3 z 4 *

5 6 z + + 5 z * z 3 * + 4 z z v * z

v * + 4 * z � * * 5

z + * 3 * + 3 *

z * + * 3 5 * z z + * + 4 4 )

3 � 4

Page 37: Goldrei.classic Set Theory for Guided Independent Study

� � � � " % " � ( � � + � ( � �

" � "

G

( � " ( (

" � � � a " ( a ( �

" � "

( � � " ( " G

" � � a " ( �

� � � " a + + ( � � " (

� ( � G � a � " % � G

a ( � + ( � � + + � G " ( � G % G � + � G �

� � ( � " (

( � ( � ( � + � � ( � G � � ( " ( � � � � � +

� ( � � " % a � % � � + �

� ( � ( " " + + " � G % � � � (

� � " ( + ( � � � " + ( � " % � � (

Page 38: Goldrei.classic Set Theory for Guided Independent Study

� � ! # ! ' � ' ) * ' # 2 4 ) 2 2 � � 2 ! ' � : 4 ' )

� 4 � ) ' � 2 # � 2 4 4 � � � 4

* : � ' ) ' # � * : * # 2 2 � * 2 ' # 2 ) 2 ' 2 4 � 4 ! � � 2

� ' ) *

2 ! 2 ) � � 2 ! 2 : 2 2 ! : * 2 ' ) * 4 4 ) ) 4 * ! 4 � 2 : 4 * 4 ! ' # 4 !

2 ) * � ' 2 � � 4 � 4 4 2 4 ) ) 4 : � 4 2 ' � * ) 4

� � ) : � 4 2 � 2 4 4 4 ! 2 ) 2 4 4 * � 4 ! 2 � ' � 4 !

� ' � � ! ! : # 4 4 * 4 ! '

� ! � : ' ) ) * * * 4 ' : ! : � 2 ! : * ' 4 ! ! 2

) * ' � : * 4 2 ! 2 * � 4 4 2 4 ! 2

2 � ' 2 2 � � � : 4 # ' ) * : 2 ! * 4 # 2 2 ! * � ' � ' 2 2 2 ! *

2 * ' ) ) 2 � � # � ! * 4 � 2 2 � 4 ' � ! ' ) # 2 ! *

' # � 2 4 ! ! ' � � : 4 � 2 ! ) 2 : 4 4 ! 2 � 4 ! 2 ! 2 # �

� � 2 2 � 2 2 ! * � 4 � * # * � � 4 # : * # 4 � ' ) 2 4 2 ! '

� 4 # # � * � 4 ! 2 2 * 2 � � � 2 � 2 2 ' : ! 4 ' � ! ' )

! * � � ' : 4 4 # * � ' # 4 ) 2 ! 2 * 2 2

� 2 : # 2 ! 4 * 2 ! 2 � ' 2 2

! 4 2 ! ! * 2 # � # 2 ' � : # � 2 ! ' * # ) 2 4 2 ' # ' * 2 2 2 * 4 ) 2

� : ' * � ) ' ) # 2 ! 2 ) ' � 4 # � 2 � 2 2 : 2 * � � ' � 4 # 2 ' #

2 ! * * ' ' 4 # 4 ) #

# ' # 2 : ) 2 # � ' 4 : : # * 4

! ' * ' ) ) ! : :

2 ' 4 � 4 � ' � * ' # 2 ' ) # ! ) ' � 4 : # � 2 # 2 ! * 2

� � ' * � 2 # 2 : * ' # 2 ' ) # 2 ! ) ' � 4 # � 2 * � 4 ' ) )

: * ' � ! ' ! *

Page 39: Goldrei.classic Set Theory for Guided Independent Study

$ % ( * + + * / / $ % 2 * ( 2 * $ 2 +

2 / ; % * + 2 @ + * $ ; ( % / $

( / % * * + ; + * /

* / % / * + % / @ ( * + + * / /

* ; ( $ 2 * 2 * ( ( @ 2 @ / + $ / $ % * $

$ % q + 2 q + * 2 * + @ 2 /

* + (

$ 2 2 % 2 2 ( $ / % * $ % % * $ $ % @ + +

* % 2 2 / @ ; * ( / $ * 2 $ ( * + @ + * / / @ /

% 2 + % * $ * 2 @ + 2 q * / $ ; $ @ 2 (

$ % * * $ * $

/ $ % / ; 2 q $ ( / q $ q / % q $ @ q + + q

* q +

* ; @ 2 / % @

(

( @ 2 * + $ % @ ( $ /

% + q ( ( * $ $ % 2 2 $ ( / / * $ / % $ @ 2 / ; q $ @ * /

( $ 2 @ / $ @ / $ % @ $ * / q q * 2 2 q $ % @ q q

% ( / / $ ( ; % ( % ( / % % q ; / $ % 2 * * + / $ % / $

2 * + / / $ 2 $ 2 q / ( + 2 @ $ / / ; $ * + / @ $ ; / q % ( * $ @ 2

@ q $ $ % * $ 2 q / $ 2 $ q / + $ 2 * $ ( ; ; $ q $ / * +

+ / / q * $ @ + ; * 2 / * / / + q / $ % / q

$ ( + / $ * $ q * * + 2 @ @ 2 (

/ % ( $ * $ / / $ ; $ + * $ 2 * + 2 / $ % * $

* 2 @ $ % * 2 * 2 / % / $ q + % * % * $ * $ % 2 * / /

/ ; $ / ; ; $ q 2 $ * $ $ +

Page 40: Goldrei.classic Set Theory for Guided Independent Study

� " $ ' � ) * - ' ) 1 1 3 ' * * $ * " ) 7 3 - " ' 1

� " * $ 1 ) 7 " ' � - " ) * $ ) 7 ' ) - 3 ' * " * 1 * - 3 * - - " '

) � ' ) 3 1 ' * " * - " 1 * - ' 3 " ) 3 " '

* � $ � ) * " ' � ) 7 7 3 * $ * � $ ' ' ' 1 " * $

* � * " 7 3 ' * " ' 1 ' * '

* ' 3 ' 1 * ' ' * � ) * ' * 1 3 ' * " * 1 3 ' 3 ) " * $ 7 1 '

3 7 $ * ' - * ' - ) $ * ) 7 ) * 1 * � - - * 1 * " - ) 7 ' � '

' ) ' ) ' 7 7 ) ' * � - 3 ) " ) 7 3 ' " 7 $ - " 1 " ' 1 ) - ) 3

" - ' ) * " 7 ' $ * - ' 1 ' * ' * - 3 - ) ) ' $ 3 7 3 ) * $

" " * � 1 $ 3 ' * ' * � ' ) $ 7 7 1 - " - " - ' * 1 - "

1 7 " * � - ) ' $ � $ � - $ � * 7 - $ * ) * 7 ' - ' * '

� " 1 ' * " * $ 3 ) 7 3 * - ' 1 " * 1 3 ) 7 ' " ) ) 7 ' - " '

* " ' * - " " ) ' 1 3 ) 7 - 7 - ) 1 ' $ $ 7 $ ) " " 1 - "

* � " ' ' � " 1 $ 3 ' * - 3 * ' * - 3 "

' ' * $ ' * " * - 1 $ * * " * 7 ' - $ ' � 7 - " ) ' �

) " 1 ) 7 7 * ' ' * ) ) " " * $ ' * � - -

) ) 7 7 $ * * ) ) " " - - ' ) * � 1 * � - ) 1 " )

* � - * * � ' ' * ' * 1 1 " * � * - * � 1 3 1 3 "

3 * " $ $ ' 7 ' ) $ 7 ) " ) 3 * � - 7 - * 1 * - 1 " * ' � $ �

' - 3 ) 3 3 * 1 ' 1 " " 3 - * * - * " "

* ) 3 ) ) * - $ ) * 1 " ' � ) * - " ' * ) * - ' � ) 7 7 � - * 1 ' * ) " *

* - " ' ' " ) 7 $ $ 3 ' - * * � " ' 1 * � ) * * ' 1 " * ' 1 * � * � - ' - ) 7 - * '

� � ' � ) 7 7 * ) ) ' � - 3 ) * " ) 7 - " ' ) 3 $ 7 7 ) 7 ' 1 1 $ - ) " ) - 1

' * ) 7 � $ * � - $ " ) $ 7 $ ) " " - " * $ - '

1 $ * � " $ � ) * - * � - " - ' ) $ " - * " 1 � * 1 - 7 ) 3 $

* - * " ) 7 " ' ) - 1 ' 1 " - ) 7 7

1 7 * $ ' - ' ' 3 * 7 7 ' * 3 ' * " * " ) ' ' * � - ) ' - * � * �

' " 1 7 - * 1 " - " - 7 ) 3 * 1 ' - * ' ' ) * ' * � - " 1 " * $ '

' � 1 $ ' 1 " � � 1 ' - " - - ' - - 1 ) ) " - ) ' 7 7 1

- ' " ' ) ' *

' ' - ) ' - $ ) 7 7 - - 3 * ) * 1 3

� * � ) * * � - 1 7 7 1 $ 3 ) 7 ' 1 1 7

* * $ 1 $ 1 3 - 1 3 - $ 1 ) 7 7 $ * - 3

" 7 7

1 " 7 7 ' * ' 1 3 * ) ' ) 3 1 3 * ) 3 ' - -

� $ 7 7

* ) * 1 - * � - - ) 3 * " " - * 3

- * 1 - 3 * 1 3 * - 3 $ * ' 7 - ) " * ' ) * $ ' - ' ) 7 7

Page 41: Goldrei.classic Set Theory for Guided Independent Study

+ + . . 1 5 . 5 5 6 9 <

E F E E 9 . 1 5 6 N 1 9 N N < 1 1 E 9 . N < N . 5 6 +

1

9 . < < N 1 . 9 + 5 6 . 1 9 E 9 5

6 9 . 6 + 9 + < 5 6 1 E 9 5

6 . 5 5 6 E + 5 + 5 + + F + + 5 1 . F 9 E + 9 5 1 N F + 9

1 E F E 5 F E 9 F + E +

9 E + 1 . 5 + E + E E 5 + < + + F 1 E E + E F

E 1 1 5 E +

6 5 6 + E + 9 5

1 . 5 E 9 9 + 1 E 5 + 9 E + 9 5 + < E 1 5

1 . F E . 9 1 N 6 1 + + 1 E E + < E F 1 5 . F + 5 6 +

9 + + 9 5 E 5 6 + 5 . F . . 5 6 + N E + 9 . 5 N E F 1 E < + 5 < E F + N + 5

9 1 N E F 5 6 5 . 9 < < N + + F . F 5 . < + E 9 + E N + 5

6 E < 5 1 < < 1 + + E + F E F 1 5 9 F 9 E E + 9 N E F E F

1 1 + 5 E 5 6 . 5 5 < < + E E + F + + < E +

6 + E < < E N 5 6 E 9 + N 1 F + < + E + E E 6 + N E 1

E 1 + 5 6 5 5 6 1 + 1 . 5 N + + E 1 N E 6 + F + + 9

E + 9 6 . N E E +

+ 1 . < < + < E N 5 + N F E F 9 N N < + 5 9 . < 9 E 5

E 1 9 + + + + . E

E 9 E 9 1 E +

6 + + 5 E F F N 5 + + F N E F

N 1 9 + 9 E + 5 6 + 9 N E 6 + F 5 E F 9 N N < +

. F 5 . 5 + + N 1 E 9 E +

N + . 1 < 9 9 E 9 E 6 + N E 1 . E < < N

+ F + E N < < E + 5 + . 5 5 E F +

E F +

+ 1 6 . < E 6 E E F 9 5 . 9 N 5 E + 9 5 N E F < + . N 5 N E F . 5 + N F

5 + 5 + 9 E + N + E E < < < 5 + + F 5 E F E F N E F

F 5 6 + E F 5 E . + 1 . 5 1 1 N E 6 N 1

Page 42: Goldrei.classic Set Theory for Guided Independent Study

� � � � � $ � $ + , $ � �

+ 6 7 , ,

, P

6 + P P � � � � + � , � + + 6 �

, + P 7 � + , + 6 � , + $ P 7 � + + ,

� , � 7 + 7 7 6 � + 6 � 6 6 , � 7

� � + + + 6 �

$ + 6 � � � � � + + � , � $ + , $ $

P 7 + , + $ $ � $ P P � � , $ , $ � � + � � + � � �

� � , � � � � , 7 , �

+ � � + � , � , + � $ $ 7 $ + ,

� � � $ $ � � P 6 $ 6 P

+ � $ + � , � 6 +

P � 7 7

, + P � � 7 7 � , , 7 + � , + � $ + , 6 + ,

$ � P � 7 P � � $ + + , $ $

� 6 � 7 � P , , , + 6 � � + , P $ + � 7 + � 7 � � � � 7

� , P 6 � P , , , P + , $ 7 + � � 6 , P + , � ,

+ , � + � � � � � $ � $ , � � , ,

+ � + � � 7 7 $ + + � , $

� + � � + � , , � $ � � $ 7 $

$ , $ � 7 P 7 � � + , $

� , � $ + , $ � 6 � +

� � + $

, 7

Page 43: Goldrei.classic Set Theory for Guided Independent Study

� � � " � " - " 4

E - � "

- E

� " � � � � - � � " - � � 4 E �

E " � � � � 4 � � � E � � E � �

� � E " " � � � E " � �

E � � - 4 � �

� 4 " � � -

� - �

4 � � �

� � � � � � E � � 4 E � E � � � E � � " 4 � - � E �

� E " � � E E " E � � � � � � E " � �

� E " " � E � " � � E � � E � E �

� E � � 4 � � E E 4 � 4

E � � E " E E � � �

� � � � � E " � " �

E " - � � � " 4 � � - � � �

E � E � E " " � � 4 � E �

� � - � � " " - � �

E � " E " E � � " � � � �

E E � � E " � � 4 � � � -

� E � � E " � E � -

� E � E " � E E � � E " � � 4 �

E � � "

E E " � � � E

E " � "

� � � � � �

Page 44: Goldrei.classic Set Theory for Guided Independent Study

� � 1 2

: ; � < = �

C C : � K � K K C 2 < = � < = �

K

� : < � < = � : ; ; : : < = 1 1 : 2 2 K C K d K 1 2

C � : < � < = � � 2 : � ; 2 d 2

; : ;

1 1: = ;

< 2 = : 2 ; ; : 2 � = � d ;

� < = � < 2 C : : ; 2 : = � 2 d

� < 2

: = : C 2 C 2 : � 2 d K ; d C d 1 2 ; C

= : ; : ; d � ; ; 1 2 � � 2 ; ; � = � = ;

; < = 2 : 1 2 � < = = d 2 K = � K d ; = ; : = C 2

� 1 1 2 : 1 2 � : C 2 C d � ; � : : 2 ; 2 d d ; d 2 : �

: = 1 1 2 : = � = : � ; � C = 1 d 2 : d � 2 C ; ; � K ; C

= : < ; � < d C : K K 1 1 2 C : K 1 2 : ; C : : 2 � : : �

� ; : � < C 1 2 : K C ; � = 1 2 � 2

C = 1 = K = � : < C : � ; � < 2 : = : C ; : 2 2 2 C = = � <

= = < 1 < 2 C 2 2 K � < 2 = � C = 1

C = K < C K K � � d 2 K � : : 2 : ; C 2

C : d = � = : � < 2 : 2 � < 2 C 2

K K � < 2 2 = 1 1 : 2 � : ; d � = K d d 2 1 2 2 d � : � 2 � : 2 2 K � < 2

; 1 < : � 2 C = 1 � < 2 2 � C 2 C 2 2 d � d � 2 d = � C = 1 = 1 : ;

; d : : : ; = 1 1 : � : � < 1 2 d : K C 1 = 2 d ; d : ; : d

; C = 1 1 1 1 : K 1 K 2 = = ; � : 2 � : � ; 2 2 d � 2 ;

< = : d 2 K : 2 : = C : K 2 2 d K � ; K d 2 d 2 C = 1

; C = 1 1 � < K : � 1 ; K � = � K ; d ; 2 = � = : � 1

1 �

� < C � : = C = 1 1 � 2

d K � : : 2 : ;

� C K : � : = d

= d = : 2 2 1 2 d �

= d d � ; ; d � ; = 1 1 : 2 � 2 2 2 ; 2 2 2 d : 2

: 2 : 1 � < : : � ; = 1 1 : : K � ; ; C 2 2 1 2 �

1 : : = < = = = C = 2 : 2 � C = � = : : = 1 = � ; =

� C � ; = : < = 1 1 K : : : � < � � 2 C : : = : = 2

: K � d 1 K < � C 1 < K : C � < 2 K � ; : d

Page 45: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � ! ! � � � 2 2 3 �

� � � � � � ! � � 2 � � ! � � � � ! � 2 � �

� 3 L � � ! � � � ! � L � � � � 3 � � 3 � ! � � �

2 � � 3 � � L � L � � � � 3 � � � � ! � � ! � � � � � � � !

� 2 � � � L � ! � � � � � 3 � � � � � ! � � � �

L � L � 3 � ! � � 2 � � � � 2 � � ! � � � � � � � 3 � � � � ! �

� 3 � � 2 2 3 � � � � ! � ! 2 � � � � � � ! � � � � � � ! 2 � 2 �

2 � 3 � 2 ! � � � ! � ! � � 3 � � � � � 2 � � � � � ! �

� L � 3 � � L 3 L � � � 3 L � 2 2 � ! � ! � � � � � ! � � � � ! � � � ! �

� � � � 2 � ! � � � 3 3 � � � ! 2 � � � � ! � � 2 2 � � � 3 ! �

� L � 3 � 3 � � � � � � � ! 3 � � ! � � � ! � !

� 3 � ! � � � � 3 � 2 2 � � ! � ! � � � � � � 3 2 2 �

3 � � 3 � � � ! � � � ! � 3 � � � � � 3 � � 2 � L ! �

� � � � � 3 � � ! � � 2 � � 3 � 2 �

� 2 � � L � � 3 � 2 � ! � � � � � 2 � 2 � � ! � � ! � � � ! � �

3 � 3 3 � � � � � ! � � � � � � � 3 2 � ! � � ! 3 � 3

! � � � � � 3 L � 3 � � � � � � ! ! � 2 ! � � � �

� ! 3 � � � ! � � � 3 � � � � ! � � � 3

� � � 3 � � � � � � ! � � �

� � � 3 L � � � 2 � � � ! � � � � ! �

� � 2 � � ! � � � 3 � � � � 2 2 2 2 � 2

� � � 3 L 3 � 2 � � ! � � � � � ! � � �

� 2 � � 3 � � 3 2 2 � � � 2 �

3 � � � � � ! � � � � � � ! � � � � � � � !

� ! � � 2 2 � 3 � � � � � L � �

� � L � � � 3 2 � 3 � ! � ! � � � � ! ! � � 2 � � � � �

� � 3 2 2 � � � � � � 3 � ! � � L ! � � 2 ! � � ! �

3 � 3 � ! � � ! 3 � 2 2 �

� ! � 3 � � � � � � �

� � 3 � ! � � � � ! � � ! � � � ! � ! � � � 2 2 !

� 2 � � 3 � � � � ! � 2 ! � � � !

� � � � � L � � � � ! � � 2 � ! � � � � � � ! � � 3 � �

Page 46: Goldrei.classic Set Theory for Guided Independent Study

+ , / 0 3 / 5 / 5 < , , 3 0 < + , 3

, 5 / 5 0 K , + 5 < < / , , K < P / 0 < 0 < < / , 3 < P / / 3 /

, 3 + , K < 0 / , 3 < 5 / < , + 5 < / < K /

5 < 3 , 5 ,

+ 5 + K < < , + + < / , P 3 5 5 , 0 < 5 , / P < 0

< 0 < , < K 0 0 5 / , , 3 K 3 , , 3 + , P 3 , / 3 , < < ,

P < / P , 3 , + 3 , 3 5 < 3 < , < / P 5 5 5 / 5 < , / 5

< P 5 , 0 5 < / , 3 P / , 5 / < 5 < 5 0 0 K 3 / , P < P 3 / P , P 0 0

5 0 0 3 , 3 3 + < , K < < 0 5 P 3 / P P 3 5 P 5 / 3 , P 3 / P / ,

, < 3 + 5 3 / < , , , P , 3 / < < , P K 0 0 3

3 P / 0

K < < , / +

K 3 , K 5 , 5 / 5 + < 3 , P / P < , < , < P + P / , < < , P /

5 0 , P

< / < < , 3 K , , / , 5 < / + K

5 5 + < , P P 0 3 3 3 3 , 3

/ + < 3 / / 3

5 / + < P / < 3 3 / / <

< 5 < 3 P / , P P , < 3 P 3

/ , < + < 3 3 5 0 0 / , P < , 3 , , P 0 0 < < 5 0 0 , P ,

K + 3 0 P , 5 < < < P / P < < , P / + , 0 5 <

5 P P 5 5 3 /

3 P , K < < P , < / 3 P / , < 3 3 3 < P , < , P /

< , K < 3 + < + P < , 0 0 3 3 3 , < 3

P / , + < , / 3 0 0 / P , 3

< 3 P / + P 0 5 P 3 P / < < P

0 3 5 < , 3 3 P , P , 3

+ P < P < / 3 5 P 3 + P P ,

3 < P < / < , P , < , P 3 3 0 0 P / , P

, 3 < / < , < + 3 P , 3 3

Page 47: Goldrei.classic Set Theory for Guided Independent Study

! # & ' ) + 0 2 ' 3 ! 2 ' ! 2 # # 0 '

#

0 & ' 2 0 ! #

# & + 2 q ' !

2 0 # ! ) ' ! # ! q

& ) & + 0 ! 0

! ' 3 & ' & # 0 & 2 ! # 2 # 0 & q & ! & ! # q ' # ' & !

' 3 & & 0 ' 0 ! 0 # ) q ' # q # q ! ) ! 2 ) ! # ! & q

q 0 3 ' ' 3 ! # ! 0 3 # 3 2

3 ' 0 2 '

3 ) # # 2 0 ! ) ) # ! # + 2 # ' # ) 2 ! 2 0 ! + # q 0

' # & q 2 2 ! 0 ! ! q ! ' # & + + q ' 2 ! 0 ! ' # & q ! ' # ' ! 2 )

q ! ) ! 2 # # 3 ' 0 # 3 & # & + 2 q ' 0 0 ! ' 0

# ! & 3 0 3 2 2 2 ! & + 0 # ! # q # 3 ! + # q ' #

3 2 ' ) 2 # 3 0 ! ! ! q 2 q 2 !

0 0 2 & # q 3 ! 0 0 # ) q ' 2 # q

3 ! # # q 3 ! q ' #

! 0 3 & + + 2 # q 2 0 ) # ! & ' ! # & q 2 ! 2 ! # )

# ) 2 ! q ! ) + 0 3 + + 0 ! + q ! ) 0 q + ' !

+ # ) + ! q #

& q 0 3 3 & ' 0 & ' ! 0 3 ! 2 q ' 2 ! ) 2 q + ! # ! # 0 2 # 0

# ) + 0 0 ! 0 # q ' & 2 q q # & 2 q ! ! )

3 2 & + #

Page 48: Goldrei.classic Set Theory for Guided Independent Study

! $ ! ' + + 1 $ 3 1 3 + 1 5 6 5 ' 8 8 1 + 1 3 +

! $ 5 < ' ! + 5 5 5 + $ ! + 5

$ 5 + $ 3 ' 1 ! 5 < ' 8 5 + 3 < < 5 5 ' 5 + + $ 3 ' < ' 8

< ! 5 ! 8 8 8 3 5 6 < 3 ! 8 + ' 5 + $ 5 ! < 5 5 ! 3 8 5 + 3

1 5 8 5 < ! 8 + ! 5 + + 5 < 3 1 + $ 1 ! 5 5 + $ 3 ! 3 3 ! 8

5 5 6 + 5 8 8 8 5 ! + 3 ' 8 8

$ < 3 ' < 5 + + 1 6 1 + 3 5 $ ' 8 5 5 8 ! + +

' ! ! + $ ! 8 8 + 3 + 5 6 + ' 1 ' 6 8 5 3 < + $ 3 ! ! ! 5 <

+ 1 5

+ $ 8 3 5 8 5 5 6 8

$ + 3 ! 5 6 ! ! 5 6 + 3 8 ' 6 8 5 $ + ! ' $ 3

1 + $ ! 5 ! 6 + $ 3 3 3 < ! ! 1 5 6 ! ' 6 1 6 5 < ' 6 ! 5 5 6 ' 8

5 1 ' + 3 ! < + 1 5 + $ 3 3 ' 6 3 ' ! 3 ' 3 + 5 1 6 1 + 3

! 3 ! + 5 + 5 3 5 3 ' 3 3 8 8 < + 5 5 1 3 < 3 5 ! ! 8 3

6 ! < + 1 5 ! 5 ' $ ' ' 8 3 3 6 + 5 $ 3 ' ! 3 $ 3

' ! ' 5 5 < 1 + 1 5 6 + 3 + 1 5 6 5 3 + 1 5 5 ! $ 6

1 ! 5 $ 1 $ $ 3 5 8 ' 6 5 3 5 3 ' $ < + $

' 1 +

5 + $ < + + 1 6 1 + ! + $ ' 5 + 5 8 3 3 3 8 3 3 + 5 ' 6 ' + < ' 8 6 3 <

' 8 ' 6 < ' 8 6 < + ' 8 5 3 $ < ' 8 6 < 1 ! $ 3 3 5 ' 8 8

< 3 ! 5 < !

' !

5 5 + $ 5 < 3 + 3 + 5 + 1 ! 6 + 1 + 3 8 3 8 3 3 !

1 + $ ' ' < 3 < 5 3 6 3 $ 3 6 ' + ' 8 6 < ! $ 1 8 + + 3 5 3 ' +

1 5 8 ! 8 1 5 3 < + $ 3 1 + 1 + 5 6 + ' + 3 ' $ ' ' 8 3 < 1 ! 3

! 5 ' 8 3 ! 5 ! ! 5 6 3 $ ' 3 ' 6 5 ' 5 6 ' 1 5 6

1 5 < 5 6 $ ' + ' + + 3 5 3 + < 3

5 ' 3 < 1 + 3 3 6

3 < + 6 5 <

3 ! $ ' 8 8 5 3 + $ + 1 ! ' 8 3 ' < 5 3 5 1 ! 5 < 5 1 + ! 5 6

< 1 $ + + 1 8 8 ' 8 ! 5 8 ! ! $ ! $ 5 6 + $ ' + ! ' + 3 5 ! 5 !

Page 49: Goldrei.classic Set Theory for Guided Independent Study

! " # # ) # , 1 " 2 # # 5 5 8 8 # 2 8

5 2 # 2 # " 2 5

, # 8 ) " 8 ! # 8 5 # 2 1 2 , ) , 5 8 8 5 ! 5 2 # " 2

5 # 8

5 ) 2 !

5 # 8 5

1 8 8 5 2 # 5 5 ! ) 2 1 8 2 ) ! 5 2 ) # 2 2 # 2 5 # 5 5

" " 5 2 ! " ! 2 # 8 5 # 2 2 2 5 1 2 1 5 ! " ! 2

1 ! 5 " ! 2 ! 2 ) 8 " ) ! 2 5 # 2 1 8 "

2 ! 5 ) 8 ) # # 5 2 # " 2 5 2 # 2 )

5 5

) 2 " , 2 # 2 8 8 " 5 1 ! 5 ! " ) ! 2 # 8 # 5 5 # 8

) " # ) ! 2 5 # 5

! " 5 ! ! " " " ) ! 2 5 # 8 8

" 8 ) # # 5 5 " 5 #

# 2 5 5 " ) ! " ! 5

! " 5 ! # ! 5 5 #

2

2 !

) , ) 8 5 " ! 2

# ! ) ! ! 5 ! ! 5 " 2 #

5 " ) " 2 ) ! #

1 " 5

! ! 8

, " 5 "

1 2 2 5 5 8 1 5

2 2 2 " , ! 5 " "

, ) 2 5 ) " # 2 )

2 , 2 ) 2 5 2 # " 1

" ) 1 ) 5 ! ) # 5 5

)

! 5 5 # 5

# ) ! !

2 ! 2 8 2 " ) 5 " ! 8 8 5 ! " 5

Page 50: Goldrei.classic Set Theory for Guided Independent Study

' * ' + * - . - 0 1 1 - 1 5 - * . * J . 1

J 5 5 . ' 5 T J + * 0 + J . - 5 1 0 * 5 5 J - J 1 1

+ . * 5 * ' 1 0 1 J *

' . *

. * . + 5 + - J 5 1 T 1 1 J * . 1 + - J * - T * +

- 5 * + 1 T 1 1 * - 1 .

5 5 . 1 ' + * + . ' 5 + - 1 + T T

. 5 . 1 * + + J * 5 1 0 - 5 J + * - * 1 J

' - * 5 5 . T * * 5

* 1 0 . * 1 * + * 1 1

- * 0 + . - * 1 5 5 * * 5 J * 0 -

J * ' 1 5 * ' 0 + . 1 * +

+ + 5 - * - J +

J 5 * 1 5 - - 5 J 5 * - -

+ - J 5 -

- 0 1 1 1 5 J

1 5 0 1 1 + T 1 1 T J 1 0 0 5 * 5 5 J 5 1 0 1 T + 5 *

* ' 1 . 1 . 5 0 J 5 J 5 . 1 5 5 . ' + T T + 1 1 1 - 1 5 5 * 5 -

5 * 1 - J - - J - 5 J . 1 *

5 0 T

- - * 1

*

0 5 ' 5 0 - J * . - . 1 . 0 + * 5 + 0 ' 1 0 5 1 - - . - J

J * 5 * . ' T 1 J 1 5 . . T 1 * 5 5 1 - - * 5 * J 1 1 5 . 1

J * 5 . 5 T * 5 J 5 - 1 . * 1 0 1 5 * * * + *

0 1 5 J 1 5 + T T

Page 51: Goldrei.classic Set Theory for Guided Independent Study

� � � � � & ' ( & � � ,

( & & & � ' E ( � � & , , ' ( ( & ' O O

( � O O ' & � � ( � ] & ] & ' E ] , ( � E O O ' , ( , ] �

� � , � & ' & & � E & ' E O � ' E � � , E � E O ( O E � � , ' � E ]

( � & � � E � & ( ( , � & ( � , ( � E O O & ' � � ' & ' �

& E O ' ( E � � E �

& O � ( � , ' &

' & � � � � & O & ( � E � & , ] ' E O ] , ' , ' ,

& ' � � , ( ( �

� � , ]

� � , ' ' & ] � ] ' ,

� ,

� ' ( E , ' E ] '

& ' � ( O , ( ( � E �

� �

( E & ( O , , &

& ( , � � E & O �

� ,

] � � E � , E O & � E � � , ] � E O ( �

� � E & ( E ] , ,

& � E � � � & '

� � � � � � � & � , ( ( ( � E � (

� � ( � � E , � � E �

& � � , ' ( ' , ' & ( � & � �

� & O � , & O O & ( � & � , & ' , O O

( � & , �

( ] ] , ' , � O � & � ] & & ( , � �

� ( E ' ' � � ] E (

( & � � E � � � ( E ] , O , , ] � & ( E ] , O , � &

� ' E ' , � & & ( � O � � � E ]

Page 52: Goldrei.classic Set Theory for Guided Independent Study

� " # " � � 1 2 6

6 � # # � F 6 " 6 # � " # 1 # " # F

� F U " 1 U # � X " 6 " # # � 6 F 2 2 6 U #

6 2 " 1 " 2 # 6 " F 1 1 2 X " U 1 e F � 1 2 X

6 � 1 # 6 " # 6 " 1 1 � 6 1 2 " 1 X X F

� " 2 # 6 1 2 # 1 2 " 6 " # X 1 2

� X U # " # � 6 6 1 2 # 1 #

F X 1 F 1

6 � " # # � 1 # � " # 6 � 1 2 X 1

� 1 6 6 # F 1 " 1

1 6 � 1 1 � " e 1 � " � 2 # 1 1 U # � 2 1 1 1 � X 2 X

2 1 � 6 U F " 1

" � 1 # � " 2 " U 1 2 1 2 # X

# 1 1 1 2

1 2 � 2 # � # 1 2 U 1 # " 1 X � 1 U 6

" 6 � U U 6 F # 1 e 1 e 1 " # 1 6 � 1 # " 1

1 1 2 " U U

1 2 " 1 � " X " 6 � X e # 1

1 � X e # X X

� 6 # � 2 6 U � 1 6 1 2

1 6 6 # # � 2 # � 1 6 1 6 1 # � " 1 2 1 2

" 6 � # � " # � 2 U # � U 6 1 2 X 1 1 2

� " 6 # � " # " 6 X � " # �

6 # # � X � " X

� # " 1 # # � 6 X # � " � 6 " 6 # � # 6 1 #

6 1 � " � X � 1 2 # 2 F 1 X # 1 � �

" U 6 " 6 6 # X 1 6 1 6 1 2 1 � F

X 6 6 1 � # 2 1 2 � e

� 1 2 F F F " # �

6 � " 1 1 " F 2

� 1 1 � 1 6 # 2 " � 1 " 6 U # 1 1

Page 53: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � ' � ' ) � � � ' � � 0 � ' 0

� � ' ' ' � � � 0 � � � � ' 0 � ? ? � � � �

? � � � ) � � � � � ' ) � �

0 � � � ) � � � ? � � � ? � � 0 � ' � ? 0 � 0 � �

� ' 0 � � 0 � � � � ? ?

� � � �

� � � � � � � 0 � � � ? ' 0 � � ? �

� � � � ? ? � � � � ? � � ? � 0 � 0

� � � � � ) 0 � � 0 � 0 � ' 0 � 0 � ' � 0 � ) �

� � 0 � � ? � ) � � � 0 �

� ? � 0 ' � � � 0

' 0 � � � �

� ) � 0 0 � � ' � � � ? � ? ? � ? �

� ? � 0 �

� � ) � � � � 0 � � � � � �

� � ? ? � � � � ' 0 ? ? � 0 � � � � ) � � � �

' � 0 � � � � ? � � 0 � �

� � � � � � � � � ?

� ? � � � � ? � � ? 0 � ) � ? 0 � ' � � ) � ?

� � � ) � � � ? � 0 � � '

� � ' 0 � ' � � � 0 � ' ) � � 0 � ' 0 �

' ) � � � ) 0 � � ? ) � � 0

' 0 � ' ? � � � � � ) ' � � � ?

' 0 � ?

� ) 0 � � ' ) � � ' � 0 � � � � ) � � � ? � � 0 �

� � � 0 ) � ) ) � � ) ' ) � � ) � � ' ) � � �

Page 54: Goldrei.classic Set Theory for Guided Independent Study

% ' ( ( 1 2

e % ' ( ( % l p q q l 2 % q 1 l % 2 2 ~

% ~ ( ( e e q p ~ q q ~ 1 2 q q q ' ( q ' ~ l q ( q q

% p p 1 q ( q l ~ 2 % q p ' q 1 ~ q 1 q 2

1 ~ 2 ~ l p e q ' 2 q 2 % 1 l p

p ( ' ~ l ( q l % q 1

2 ( q p ' q e q ' ( l 2 ' 2 % l e l % q e

% 1 q 2 q e ( ( q 1 ( ( l % 1 l % e

2 q 2 q l q 1 % 2 ( ( ( % % % % % 1 l 1

p % l p ' l % e ( ( 1 ' 2 % 2 1 1 ~ e % q %

p % p ' % ' ( ( % 2 2 l ( 1 q % e % 2

2 q ( ~

% p ' l % ' ( ( l q 2

p % l p ' % ( ( 1 2

% ' l % e ( (

% ' % e ' ( ( 1 2

l q ~ q l q ( ( % e q e q % ' ( % ( 1 2 q ( % e q ' q ' % e

% ~ q ' l p ( q ' ~ l q ( q q 2 ' p ' l p q e q e % q e ' %

l p q

1 ~ 2 % 2 q l % e ( q l q q q q %

% p ' ~ 2 l 1

Page 55: Goldrei.classic Set Theory for Guided Independent Study

4 5 6 8 9 8 5 8 8 N O

9 8 N N O O 5 6 8

u u O 4 9 6 8 O

u « 6 O O ½ 6 ½ «

u 5 8 8 8 8 4 5 8 8 8 « 8

N 8 N O 9 ½ 8 N 8 O N 8 4

O 4 8 O 4 O « O ½ N ½ 8 N

O 6 O 8 « « 8 N 9 O 5 8 O N ½ 6 ½ N O 8 N 5 ½ N O

½ O 8 4 O

6 8 N 5 N 6 ½ 8 N 8 N 5 4 9

u O N O O ½ 9 9 5 N O N 8 N 9

6 u 5

4 5 « «

½ 4 8 5 « « O 9 « « O N O 5 ½ 6 8 N O 4 ½ 8 O 4 O 4 9 ½ 8 N 8 O N O

4 5 O ½ u O 8 4 O ½ « « O O O 9 8 6 5

½ 9 8 O 4 9 8 O u 5 9 ½ O 6 u 9 O 5 « 4 O O 4 8 u ½ 9

O u 8 8 8 ½ O 9 O 9 ½ O 5 « O 8 ½ u ½ 9 9 O « 8

6 O 5 « 8 8 6 O u 8 O ½ 5 9 O 9 6 O

« O 8 8 8 N 8 4 ½ « N 6 « 8

4 5

O N O 8

4 N O 6 ½ 8 ½ 9 6 6 8 4 N 5 6 4 5 « N O 6 9 9 8 6 8 O

6 u O 5 O 8 8 O ½ 8 N 8 8 « ½ O 5 9 6

5 O 9 u ½ u ½ 4 N 8 u 8 5 « u u

5 9 O O « ½ 6 6 ½ 9 O ½ O 6 4 O 8 6 8 N O 6 8

O ½ O O N O 5

u 9 O N O

N ½ 8 5 N 5 O 6 8 8 4 N ½ 6 ½ 8 N 8 O u 5 8 8 N «

N ½ 8 N O 5 O u O « O N 8 8 N 5 O « 6 O O 8

6 ½ 8 N 8 N ½ 6 « 8 O N ½ 5 8 4 9 8 N

6 O ½ N N O 6 O 4 8 N N ½ 8 5 « u 8 O ½ 8 4 6 N

8 6 u 9 ½ u N 8 O 8 ½ 9 4 8 O 8 ½ N 8 8 4 O O 5 4 N ½ N

6 u 5 O 5 4 9 O O ½ 8 O O O N 6 N 5 4 5 « N O 4 9 9 O 5 N 6 O 8 4 O

« O 4 8 8 N O 4 4 O 8 4 O 5 « ½ 8 6 ½ « «

Page 56: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � % � � � � % � � � � � � �

� � � � � 5 � � � � � � � 5 � � � � � � � � � � � � � �

� � % � � � � 5 � % � � � % % % % % � � � � � � � � �

� � � � � � � � � � � 5 � � � 5 � � � � � 5 � � �

� � � � � � � % � � � � � % % � � � � � � � � �

� � � � � 5 � � 5 � % % % � � � �

� � � � % % � � � � � � � � � � �

� 5 � � 5 �

� � % � � % % � 5 � � � � � � 5 � % � � 5 � � � � �

� � 5 � � � � 5 � � � � � � �

� � � � � � � � � � � 5 5 � � � � � � � � �

� %

� 5 � 5 � � � � � � � % % � �

� � � � � � � � � � 5 � � � 5 � � � � � %

� % � � � 5 � 5 � � 5 � 5 � � � � � � � � �

� 5 � � � � � � � � � � 5 5 � �

� � � � � % % 5 � � � % � � � � � �

� � % � � � % % � � � � � � � � 5 � � � � � � � � 5 �

Page 57: Goldrei.classic Set Theory for Guided Independent Study

= > @ = @ > C D E E

X X @ ^ > _ > E @ ^ E ^ D @ > _ g = D _ E X g ^ D E g ^ _ X ^ > _ ^ _

X > C D _ X D _ D E E _ D C D E _ X C

> D D C ^ ^ X D > = X ^ _ = X X > X X D @

E E C X D E E _ X _ X

> X _ ^ _ @ > D _ @ C > g E X _

@ X X E ^ E X > E X X _ > D E D C X ^ ^

@ @ ^ X = ^ > >

= > E X g X @ > _ X = ^ ^ ^ _ @ C > D _

D _ X = _ C X C > _ g > _ _ @ > _ _ @ g >

^ > > _ g C X _ @ > _ g D X @ > _ g @ >

E D X E X

_ X C g > = X > C X D E _ > D @ > _ > C E ^ E ^ @ ^ > _ X > X g

X g ^ E D X @ @ _ X _ X _ @ ^ D ^ > _ X _

E C X _ X X = X X _ ^ > g D _ X X > ^ ^ _ @ = X X _ ^ > _

X _ g X X > X > _ X > C X _ E E @ = X > E E > ^ _

Page 58: Goldrei.classic Set Theory for Guided Independent Study

� � � � � $ � $ ( � � $ ( , � � $ , � $ 1 1 $ �

� $ � 1 B , $ E , � $ E � � , � $ E $ , � � E � ( �

, 1

� � B � $ � , $ � $ � � � 1 1 $

,

1 � B $ � $ � $ 1 , � � $ E � E � E � �

$ � � ( , $ � ( , � B $ $ , B $ E � , � � � $ � � �

E B � $ E $ 1 1 , B $ , ( � 1 � � B E E � � $ E B E $ 1 B E B , �

� � � � $ � � B 1 � � , � � � , , $ � � E � � E B E

$ , � $ � $ $ B , $ � � $ � $ 1 B , � � 1 � 1 B , � B $ E

$ , � $ E

B $

B $ $

� E B � $ $

� � B $ E $

� E B $ E $

� $ $

� B $ E $

� � � $ � $ 1 � $ � ( , � B $ B � $ � � , B $ , � � �

B � � � B $ � $ ( � � � $ � $ $ � � � � $ � � B � $ � $

� $ ( � B � , B $ � $ � B , � $ $ , $

Page 59: Goldrei.classic Set Theory for Guided Independent Study

� "

� � " "

" "

B " D "

B " D "

B � D "

� � " "

B " D "

B " D "

e f " D e " j " B f B " l l " D B � B

s

f " f f B B B j e B l s l B B " e " j B

f j B e l e " s � D B B B j B l B B B s s B B

� B j l B B f B f B " B " D B B " � D s D B

f s � � " D s e " � B � B

" l

s f

" l l

s � f j " j j e s l l l " s

f � e " e e B " " B f " l � B D B " s B B s l " "

f e " " B f B D s l � D � " D � e " l " B

j j � e � " B " f B D � " D " s D B j

e f l " l f " e " e s s " j � B e f � B B e � B " s

" f " j B " " B e f l l " j " j " e " " B "

l B B j e " " " " B B � e " " B " " e B j B

D " B e " " f B " B j � e B l " D " " s " B

Page 60: Goldrei.classic Set Theory for Guided Independent Study

) , - - ) , - , 6 - - A , A 6 C

d , g 6 C , C ,

, C 6 C 6 g , �

d A g g g , - - 6 , , A

g g ,

6 d , g 6 C g g , - C g 6 g ,

C C g , C - g C g , �

g d g g g C , È , � È g 6 C , � , È - C - g

6 C g ,

, C ) d È A , È C g È d 6 A È ) È , C g g , C g g C �

, - -

6 ) È 6 , g 6 C g g C g g , � g - È g È � È g A , È

d ) � g C g , � , - g C , � g � C È d , 6 g C g , � g C g C 6 g -

, d 6 - g ) , È A g , 6 g C g , A d È , A È 6 - g � ) d � , - g �

- C , � g È g d È È d , A , 6 - È 6 - g d A , È g È d - - g C , �

, ) , , È , È � C g g , � g È � C ) 6 C È È g C g , � , � È C g g , �

- , C d - C g - 6 g

g d C - , C ) d d , g 6 C , g C g , �

, - d - - d È g � g , � , � � È , - È g � C , C

,

g ) , 6 6 - - È C ) g C g , 6 g 6 - È

A d , C ) d È A , , g C , d C 6 C C È A ) g ) 6 g 6 - È , ) , , d d ,

C C g , C , È d 6 È A ) È A È , d 6 È ) , d ) È

g , � 6 A g - , A ) È g È C È , - 6 d C ) 6 C ) È

d - C ) , - d , � , - A C , d , È

d - C , A È - È g - - ) 6 È A , - È

A C , A - C È C ) È g È d , C È È g A È 6 C g , g C È d

, C È � , d , È C g � C ) , C È C g , �

, C , C d g � , g C , È d È -

C A C ) C , 6 � , , g C È

Page 61: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � ! � � � � � � � � � � � � �

� � � � � � � � � � ! � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ! � � � ! � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � ! � � � � � � � ! � � � � !

� � � � � � � � � � � �

� � � � � � � � � � � � � � � � ! �

� � � � � � � � � � �

� � � � �

� � � � � � � � � � � �

! � � � ! � � � � � !

� � � � � ! � � � � � � !

� � � ! � � � � � � � �

� ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � � !

� � � � � � � � � � ! � � �

� � � � � � � � � � ! � � � � � � � � � ! � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � � � !

� � � � � � � � � � � � � !

� � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! � � �

� � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ! � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � � � � � � � �

� �

Page 62: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � # $ � # � $ $ � # � � �

2 � � 3 2 � 2 � $ 3 � 2 $ 3 � ? 3 � � � ? 3 � 2 2

� ? 3 � F � � 3 � � 3 � � � � � F # � 2 $

3 � � 2 � 2 F $ � � � � # � $ 2 � � � # � 3 $ � # ? $

2 � 3 $ ? $ � 2 $ � # � $ � $ � $ 2 3 � # $ # $ � # � # $ �

� ? � 3 F � � � � # � � � $ � # ? � � � � $ $

� � 2 � $ 2 $ 3 2 � � � ? �

3 � 2 � 2 � � � 3 # � � ? $ # 2 $ # 2

� # � 2

2 F � $ # 2 $ � � 2 F � $

2 F � # � � $ � # $ 2 $ � # 2 2 � # � $ � 3

� � 2 F � F 2 $ � # 2 # 3 2 � � $ $ # # �

# � � ? ? F � 2 � 2 � ?

3 � $ # 2 $ � # � # � � � � 2 ?

� �

� 2 � 3 � 3 2 � � ?

� 3 F � �

� 3 ? 2 $

$ $ � $

� 3 � � $ � � � ? 2 � ? ? � $ # 2 $ � #

3 � � # � ? � $ 3 � $ � # � $ # 2 $ �

3 2 � � � � $ $ # � ? � � $ � $ 2 $

� � � F �

$

?

F 3 2 � ? 2 2 $ $ $ � $ ? ? $

� ? ? � � � � $ $ � $ � �

$ � � $ $ # � 2 $ � # ? ? $ # � ? 2 � � $ $ 2 $ �

� � 2 � � 3 2 � � 2 � $ � ? 2 $ $ � 2 $ � # � � $ � # � �

? � $ 2 F � $ ? � � F # � � # � � � � � 2 � 2 � 2 $ 2

Page 63: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � # � � ) � ) ) � �

) � # � � � � � � � � � ? � � � ) � � � ) � ) � � � #

#

� # � � � ) � � ? � � ? � � ) � � � # � � � � ) # �

� � # � � � # # � � ) ) � ? � � � � �

) � � � � ) � � � � � � ) )

� � � � � � � )

� #

� �

� ) � # � # � # � � � # � ) # � � � ? # # � � ) # � � � � # # ) �

� � � � � � � � ) � � # � � � � � � � )

� � � � � � � � � � # ) � � � � � ? �

� ) � # � ) ) ) � � � � � � � ) ) � �

� � � � � � � # � ) � � � � � ) � ) # � # ) �

� ) � � � � � � � � � � � � � # ) � � � � � �

� ) � � � � � # � � # � � � � �

� � � � � � � � � � � � ) � � � � � � � � �

� � � � � � � � # � � � � ) � � � � � ) � � � �

� � � � � � � � � � � # ) � � � � � � ) � ? �

� � ) � � � #

� � � � � �

� � � # � ) � � ) �

� � � � � � �

� � � � � ) � � � � )

Page 64: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � # � � � # � � � � ' �

� / � / � # � � � � � / � � � / � � � � � � � � �

� � � � # � � � / � � � E

� ' � � � � # � � � ' � � � � � � / �

' � � � � E � # � # � / � � � E

� � � ' � � � � �

� � � / � � � � � # � � � # � � � ' � ' # � # # � � � � � �

� � � � � � � � � ' � � � ' � � / # #

/ � � �

� / �

� � � � # # � � � � / # #

/ � � '

� � � �

� � � � � E

E ' / � � � ' # � ' E � � � � � � / E / � � � � � � � � ' E / � / #

E � � � ' � � � � � � � � ' � / � � / � ' � � � � / # # E � � � � �

# � ' � � ' � � ' # � � � � � � ' ' / # � � � � � ' � ' � � / � � � � ' � #

� E � � / # � � � � ' � / ' � � E � ' � / # � ' � � E � � � ' � / � # � �

� � � � ' � ' E � / � � � � ' ' � � � ' # � � � � � ' � � � � � / � � �

/ / � � � E / � / E / E � � ' ' � / E � � � � � � � / � � / # #

# � / � E � � E � � ' � �

' �

/ � � / � � / � # �

� � � � � / � �

� � � � / � � � / # # � � � � ' � � � � � � / � � E

� � � � / � � � / # # � � � ' � � � � � / �

� / E � E ' � � � � � � � � � � � � / � � � � � / � � �

� � � E � � � ' � � �

Page 65: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! " � � � � � � � � � � � 1 1 5 1 1 " !

5 � � 5 � � 5 1 1 " !

� � 5 � 5 1 1

� � � 5 ! � � 1 1

� � � ! � � 5 1 1 "

� � � � � " " " ! !

� � é � ! � ! " � �

� � � ! � � ! 5 ! 5 � � " �

� � 5 " � " � ! � " ! " � " � " � � � " � � "

5 � 5 � " � � 1 5 1 " ! � � 1 � ! �

é � ! " � " ! � " " � � " ! " ! " � "

� � � � � � � 5 1 � ! � � " � � � � "

� é 5 1 1 � � � " � � ! 5 " � " ! � " 1 � � � ! � " � �

� " " � " 1 1 5 � � " 5 1 � � � � � � ! � � � 5 ! " � � � 5 5 �

� é

! ! " 1 � " � � ! � � 5 � � � � � � � � 1 � � ! é 5 � � � �

5 � 5 1 � � � ! " 1 � � � é

! 1 � � � ! � � 1 � � ! � � 1 � � � ! ! 5 � � 5 1 1 � !

� é � ! � 5 � " " � ! � � � " " � ! !

� " ! 5 1 é � ! � � 5 ! � 1 � � é � � 5 � 1

� � é 5 é � 5 ! � ! 5 " � 1 � � � � ! " � 1 �

� � ! 5 " é � 1 � � � ! � � ! 5 " � 1 � � � � � � "

� ! é ! ! é ! 1 é � ! � 5 ! � 5 1 � � � " � 5 � " ! !

5 � � ! 5 � 5 1 � � � é � é � 1 � é 5 � 5 ! ! � é ! é � � � " ! é � 1 5 �

1 � ! ! � ! � � 1 1 � " � � " " �

Page 66: Goldrei.classic Set Theory for Guided Independent Study

� � � � & � � 0 0 8 0

& � 8 � � � 0 � � � � 0 A 0 8 A �

& � � � � L � 0 & �

� � & � � 0 � � � 8 A � 0 0 A & � A A �

� L 8 � & � A � � � � �

& 0 8 � A & A 8 � A & � 0 �

A � � � � � � � 8 � � �

& � � � & � A & A

� �

8 A

8 �

� � � A � � 0 A � A 8 0 � A � � A & � � � � &

� � A & L 0 �

0 8 A A � & � 0 0 � 0 �

0

0 A � � 0 0

0 A � 0 A

A � � � 0 � L � � � � A & 0 L A

0 8 L A A � � � & A 0 A 0 A � 0 A 8 0 & 0 � � & 0 0

� A 8 0 � � � 8 � A � � 0 � � � A L & 8 L � A � � �

A & � 0 A � � � 0 A 0 A � � � 0 �

� � A � & � � &

& � � � �

0 � � k & � A � � & A � � 0 � 0 A � � � A � &

& � 8 � � � � � & � 0 0 8 � 0 � � � � 0 A � A �

& � 0 0 � 0 A 0 0 0 & � 0 � L & 0 A A 0

0 � � A & L & � 0 � � 0 0 � 8 � A � � 0 8 A

& � 0 A 0 0 0 8 � � A L � A � 0 A 0 A 0 0 A & 0

Page 67: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � # � � # � � �

� � # � � � � M � M � � M � � � � �

� � � M � � � � � � � � � � � � � � � � � � � � � M

� � � � � � � � � � � � � � � # � � � � � � � � � � � # � � M � � � � M �

� � � � � � � �

� � � � � � � # � � � � � � � � � � � M � � # � � M # �

# � � � � � � � � � � � � � � M � � � � � � � � � M � � � M � �

� � � � # � � � � � � � M � � � M � M � � � � � � # � �

# � M � � � M � # � � � � M � � � � � � � � M � � M

� � � � � � � � � � � � # M � � � � � � � � M �

� � � � � M � � � � � � � � � � � � � � � M �

� # � � � � � � � � � � � � � # � � � � � � � � � � �

� � � � # � � � � � � � � � � � � � � � � � � � � � � � � � � � � � # � M

# # � � � � � � � � � # � � M � M � � � � � � � � � � � � M �

� � � � � � � � � # � � � � � � � � � � � � � � # � � � M � � M � �

� � � � � � � # � � � � M � # � � � � � M � � � � � # �

� � # � � � # � � � # � M � � � � � � � � M � � � M

� � � � � � � # � � � � M � � � � � � � � � � � # � � M � M � � # �

� # � � � M � � � � � � # � � M � � M � � � � � � # M � �

� � � # � � � # � � � � M � � � � � M

# � M � � � � � � � � � � � M � # M � � M �

� M � � �

� � M � � # � � � � � # � � � # � � � � � � � � # �

� � � � � � � � � � � � � � � � � � � M � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � # � � � � � � � � � � � # � M # � � � � � �

� # � � � � � � � � # � � � � � � � � � � � � � # � M � � � # � �

� # � # � � � M � � � � � � � � � � M � � � � M � � � � � �

� � # � � � � M � � � � � � � � # � # M � M � # �

� � � � � � M � � � # # � � � � �

� � � # � � � �

Page 68: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � �

- � � � � � � � � � � � � � � � � � � � � � �

� � � - � � � � � � � � � � � � � � - � � � - � � � �

� � � - � � � � � � � � - � � � � � � � �

� � � - � � �

� - � � � � � � � � � � � � � � � � � - � � - � � � � � � � � � � �

� � � - - � � � � � � � � � � � � � � � � � �

� � � � � � �

� � � � � � - � � � � � � � - � - � � � � � � � � - � � �

� � � � � - � � � � - � � - � � -

� � � � � - � � � � - � - � � � � - � - � � - - � � �

� � � � � � � - � � � � � � � � �

� � - � � � - � � � � � - � � � � � � � � - � � � � � � � �

� � � � � � � � � � � � � � � - - � � � � - � � � - � � � � � � - �

� � � � � - � � � � � � � � � � � � � � � � � � � � � � - � � �

� � � � - � � � � � � � � � � � � � - � � � - � � � � � � �

� � � � - � � � � � � � � � � � � - � � � � � � �

� � � � � � � � � � � � � � � � � � � - � � � �

� � � � � � � � � � � � � � - � � � � � � � � � � � �

- � � � � � � � � � � � � � � � � � � �

� � � - � � � � � � � � � � � � � � � � � � � � �

� � � � �

� � � � - � � - � � � � - - � � � � - �

� � � � � � � � � � � � � � � � � � �

Page 69: Goldrei.classic Set Theory for Guided Independent Study

� " # $ $ " $ * � - * / " - 2 � � # � - * * - - � / � / -

� � 2 * � = � # * $ - = � � 2 - = � / $ � � 2 * $

� - � I " � - $ � - I - $ � = " � # $ * = * / / * - # / # $ " - -

* � * - $ * � / $ - / - - $ # $ 2 I / - = / - * � / " � I

2 * I * = # * $ # I * * � * / $ $ $ - = - # 2 - * / - - - #

- $ - # / # $ # � $ # - 2 / � *

- = $ - $ * = $ � - $ " - - * = * * � / / 2 - -

# � $ 2 # * * � # $ " # $ " - * = # * - * � / " - / * =

/ = - / $ * � * = # I = � # * - $ * � / � / = * - $ " - 2 - = 2 * $ � �

$ " * = $ � $ = = - $ �

= * � / � / - = � * = - / $ - * � I * * I

* # I � � / $ � - 2 - - 2 = - / $ � � I - /

� 2 $ � # - * / / � I / - 2 - � - � - *

� $ 2 * * � � $ * = * / - - * / / -

� " # $ * " = � * " � $ I � = = � - - 2 # I

� " # " 2 * � * - * � / - # / � " $ " 2

* = # * * � / � = * -

� = $ - * - � � I - 2 * / I � I � 2 * - I * * = * - # - $ $ -

� / � / - / $ * � / 2 � $ * $ I * � $ �

- / * - - / 2 � � 2 2 - - # * � / = � / # 2 # I / 2 * *

= - $ - � = � $ 2 # $ � 2 * / # / - $ $ - � � / * $ - I * / - I

� 2 - 2 = $ = * = * / = � = - $ $ - $ " � 2 � � / $ = - $ =

= " I I = - * " # $ 2

� - $ " = " # - 2 I $ = = $ * # $ - 2 - # * � / = * - / $ # " - *

� = * - = � = = * $ - � I � * - = # � 2 * / $ # / - � - $ * �

� " * * $ $ * # 2 - 2 = - $ $ " � 2 * I I � 2 # "

= - $ # 2 � - - $ " * " - 2 * - # " - 2 � # I

" # � 2 � " = � 2 � / � � � 2 - - $ # $ * � / � � / �

� 2 - 2 # # 2 # I - = - $ # $ * $ * = # / # * - - $

# � � - � # � - $ - * � # / * $ - - $

� 2 $ " # � $ # / $ * " # � $ - �

/ * � $ � # $ " = - # = " # - - 2 � - I $ * " -

2 # I � 2 I # = � $ � � * / * I - - $ - $ = � * $

I - $ � 2 # # # # I # I I # 2 $ I # I = � #

Page 70: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � % � �

� � � ? � � � A ? � A F � ? ? A � � � N F

? � � % � � � � � N � � N � � A F � N � � A � N � � � � �

A � F A � N A � � N � % � � � � � � % � � N � � ? � � ? F � ? � F

� N � % ? � � F � � � � F � � A A � % ? � ? F A � � � ? �

� � � � ? � � F � � � � A � � � F � N ? � F ? � � � % A � � F �

� � � A � � � � � � % � � % � � � � A � � � F � � � � N ? % � � ? �

� % � A � � A � � � N ? � � A � � � � ? � % A F � � � � �

? � N A ? N A � �

A F

A � � % A � � � N � ? � A ? � � ? � F � � � � F A F ? ? � � � �

� � F F

� A ? A F A N � ? � ? F �

A � � N � ? �

? A ? � � ? � F % � ? � � ? ? � A

� % � � � � � N ? F � A F ? F F � � ? N � F

� � � ? ? � � � � F � � � A F ?

A F � � � F � A � � � A F

� N A � % � ? � � ? � F � � � � F � % � � �

� � A � � N � � � N � � � � N A F � F A ? F ? � F � F

� A �

? � ? � F �

% �

Page 71: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � $ � � � � & � & � � � - & / � - � � /

� & - $ � - & � � � : - & : � &

& � & � � - $ � $ � � � � : - $ & : � & � � & � � � � � � & /

� : : & $ - - � : & : &

� - � � � � � - � & / � - / � / &

$ � �

� / � - � & &

& � : : � � � / � & / & � � � � -

� / : $ � & � & � � � � � - $ �

$ & - $ � $ � / � � �

$ � � � : � & / & / & � & � / - � � � � � / &

� - � / & & � � � - � � � / / & - � � �

& & � / - � � & � / : � - & � � � � / � : � � � � / - /

� � & � � - � & / - & � � � & / � / � & : & �

� - � � � : - : & � � � & & : : / - � &

$ & - � - & � � : : � � & �

& : : & � /

� - � / - � : / � � � � - � � / & -

$ � � : / � � � / - - & : : - & � & � - & �

� � & � & & � /

� $ & � : � - & � &

- � / - & � � & � � � � & / & - � & / & - �

� � - & / � & / � � & & : : � � / � & / & / &

- � & & / & � � / & � & �

& � � � � � / � � : - & � � �

- � & � � � $ � � & � � / - & � �

- � $ � : � � / & � & / � � / �

- � � $ & � � - �

- - � & / $ -

& � $ � - - � � : - $ & : � & � & - $ � - - $ � � � � � � � � - � &

� $ � : � & � - � � & & / � - � - � �

- � & / � & - &

Page 72: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� D � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � D � � �

� � � � � � � � D � � � � � � � � � � � � D � � � � � �

� D � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � D � � D � � � � � � � � � �

� � � � � � � � � � � � � D � � � �

� � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � D D � � D � � � � � � � � � � � D � � �

D D � � � � � �

� � � � � � � � D D � � � � � � � � � � � � � � � � � �

� � � D � � � � �

� � � � � � � D D � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � D � � � �

� � �

� � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � D � � � � � � �

� � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � D � � � � � � � � �

Page 73: Goldrei.classic Set Theory for Guided Independent Study

" % ) * " " 1 * 5 5 % : " 1 < : 1 5 " ) @

5 @ 1 * < ) 5 " % < 1 C C 5 % : : < 5 : " C * ) " ) " 1 ) 1 : 1

) 5 * 5 < ) 1 5 " * : 1 5 * * : : 5 * : " < % < "

) % % 1 @ " * < ) 1 : : " * 1 * " : " < ) 5 5 1 @ : " @ :

< ) % " : " < < * ) 1 @ @ : C ) " 5 " 1 % " : " " C

) 1 < " @ 1 % 5 5 5 < " @ @ " 5 1 * " % 1 C % 1 " < ) ) * " % * 1 @

@ " < " ) : < ) 1 @ < ) 5 1 C 1 @ 1 : " * " 1

% : C 1 " C : < * " C " 5 1 < 1 @ C % : ) * ) 1 : " 5 5 @ @

1 % ) 1 < ) : " 5 * 1 C 1 " % < ) 5 @ " % : " < : 5 <

) * : : < % < * 1 : 5 1 : 5 * < ) 5 1 C < * ) < ) " % " : <

< ) 5 : 5 @ % 1 ) 5 % < 1 < ) 5 5 % * < ) 5 5 * : < 5 @ 1 *

) " @ % C 1 C % " @ 5 : ) 5 " < " ) 5 : 5 % " @ :

* 5 ) ) * : 5 " * C C * < * < ) ) 5 1 @ @ 5 1 @ @ 1 % < *

* ) < " % " 1 C " % " * 5 : 5 5 * < * 1 C % 5 " @ < * " C 1 1 5 1 * :

< < : 5 < < 5 % % * 1 % * @ 5 5 " C % * C " " 5 5 5 C 5 1 " : " %

: 5 < < 5 " ) * 5 : ) @ @ 5 C < * C < ) * : 1 < 5 5 5 * C 5 < " 1 " < 5 5

% " @ 5

< : @ " " 1 < 5 % C ) 1 " % % " @ 5 : 1 % " 5 5 ) 5 1 @ % 1 5

" 5 < * " 5 : * * C < ) 1 < % 1 % : * * 5

: @ 1 @ " % @ 1 " : % 5 ) 5 % " < 1 < 1 : " < 1 * : 5 1 < @

@ < ) 5 5 " * < < ) * : < " 5 5 C < % " : <

* " ) % " : " < * : " " " 5 1 1 @ @ " 5 : " % * * "

C 5 : < 1 < 1 < ) 5 5 5 @ " 5 C < " " 5 % @ " * * %

< 1 C @ @ " : : < " < * " " 5 " 5 % " @ 5 : 1 : 5 < 5 * % 5 : * "

* C " % : " % 5 * 5 C < 5 < " " @ 5 5 < 1 % " 5 : % * * 5 < :

5 < * " C C * C 5 C * C 5 C * C 1 5 5 1 : @ * *

* * * @ 5 C 5 5 * 5 C : * C 1 5 %

* @ @ : 5 @ 5 C 1 " C * 5 @ 5 C < : " : 5 : " % * C : <

1 * 5 5 C

" ) 5 < ) 5 1 5 < < ) 5 % 5 C < * " * 5 5 C * " C < ) 5 : 5 "

* < * " C 5 % 1 " % " 1 C * < * " C " 1 : 5 < * C < 5 * " * :

1 : % " % <

% 5 * 1 ) " % C % : " 5 % " 5 * " 5 < : @ * 5 * ) < " * <

C " " : : : 5 C 1 @ * ) * C * 5 : 5 < 5 % 5 C < @ 5 1 % " 5

5 < < : " 5

Page 74: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � " � # $ � � � � " � � � $ " � " � � "

� � # � � � $ � $ � � � � � � � � � � " � � " � � � $ " �

� "

" $ � �

� � � � R � � � � � � " " � �

# " � � � " " " � " " �

$ � � � � " � � R � � � " � " $

� � � � � � � � " � � � � " � � " � " � � " � � � $ � � "

" � � # � � R � � � � " � � R " � � " � �

� $ � � � � � � � � � � � � � � � " � � " � �

� � � � $ " � � � � " � " � � " � � � "

" � � � " � � � " �

� � � � " "

� � � �

� � � " " " � � R � R � � � "

R � � � � � � " � R � � � " � �

� " � R � " " � � � � � R � R � R

� $ � � � # � " � � � � " � � � � � � � � � � �

� � R � " � $ � � � "

� � � � $ � " � � � � " �

" " � " $ � � � " " R � �

� " � " � " � �

" � " �

" � � � � � " � R

" � � � � �

" � � � � � � � � R � � � " � " �

� " � " � � � � � � � � � � � � � " � � �

$ � � � � # � � � � " � � R R

� � R � � � � � � � " � R "

� � � " � � � " " � � � � "

� � � � �

R � � � R � � �

" � � � $ � � � �

� � # " � � R " � � " " � � � � � #

� � � � � R � � � " " $ � � � � � � R " � " � � � � �

$ � R � � $ � � � �

$ � �

� $ � $ � �

Page 75: Goldrei.classic Set Theory for Guided Independent Study

# $ ( * + , ( $ 0 $ , # 0 * # 0 > , ( + 0 $ , *

$ D E ( E + $ + $ D

+ $ $ (

* E > ( * + E , + ( * , D * , ( 0 0 > #

$ , $ > * ( $ $ + ( 0 E > , E E 0 > * * + , > > $ D ( (

+ $ , # $ 0 , , $ 0 > ( + * $ 0 0 * + * ( D ( , # $

$ 0 , D + * , E * # > 0 * 0 E + , 0 $ * ( * E > 0 $ # ( , > , +

0 , E D # , E * * > 0 $ , # 0 D 0 + , > > + 0 D , # , $

> , , D + > * 0

* ( +

D * + > $ , + , , E $ D # ( E > * $ E > , , * + 0 >

, E $ , + $ , , , + 0 > $ , * > 0 * + , # , D +

+ D $ D ( $ , ( $ , 0 + 0 > , $ , * $ + 0 >

* D 0

, 0 + , $ $ , * $ 0 + , $ ( $ * $ * + # + $ ( $ $

, $ , + , $ D $ * 0 > , # , * 0 > * 0

$ D 0 #

* * ( , E ( $ * , # # # ( * , #

( 0 * # $ 0 E * * ( $ D * * # > , , 0 0 ( # > >

E 0 0 $ ( + 0 0 ( , $ $ > * + D D ( , E , , #

$ D 0 + , $ D + ( E 0 $ ( D ( > ( E ( +

$ D , E + ( ( E $ 0 + * , E * 0 ( 0 ( $ ( > + $ 0 + > *

+ , E ( E D ( ( + 0 0 E * $ D $ + 0 $ D 0 , E (

( > , D E 0 , E , , E E , > $ * , # ( E $ 0 , , * # , D

+ > + 0 0 0 ( $ 0 ( > * # $ D + , , $ $ , 0 # * , # ( * # E 0 ( + # ( , # ,

, $ D E ( $ D $ * , $ 0 ( ( ( , ( 0 # $ , , ( # 0 E , 0

( ( E 0 $ ( , 0 E * + * E , 0 $ , * 0 0 , , 0 , E

E + , > , 0 # $ > 0 * E

$ D D $ * + 0 E ( + , $ + E * + # 0 D 0 ( * 0 + ( , $ 0 D , >

0 0 , , 0 + , * , E ( $ 0 # 0 # 0 E 0 * # #

E , > ( ( 0 $ * D $ 0 * 0 0 D * # ( > > , E 0 , 0 $ ,

+ E E * + * # $ D D 0 E * + * > ( ( $ 0 E $ 0 E D ( 0 0 E ( D

$ ( > * , * $ D E , D 0 + ( D 0 , * # $ , D , $ D * , , +

, + D , , $ D D + * , * ( D , ( E > ( + ( , + , $ 0 + * +

( E ( , 0 + D + 0 D ( 0 , $ , , 0 E , E , + , # ( $ * ,

, , $ $ D 0 E ( , + 0 , * 0

E $ , ( > > + $ ( 0 $ , , E * > * $ $ , ( 0 ( $ > ( ( + $ * 0 $ +

( D E > ( $ , + 0 0 0 $ D 0 # ( 0 > $ ( $ , , # 0 0 > , #

$ , ( , $ D E E 0 ( # # $ ( > > ( 0 $ , 0 , E 0 E 0 > ,

D ( $ + E ( E 0 ( # D , # 0 + $ + ( # , 0 E , # 0

D , D 0 * , E # * > D > > , + + $ , , > 0 $ + 0 $ , $ D 0 E , # $ D 0

+ + , D ( E * ( , , 0 E 0 0 + # 0 # ( > ( # $ + # 0 0 + $ ,

Page 76: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! " � & ' ' " � " � � � - � � ' ' �

� ' � 2 " � � " � � ' - - - ! - � " � & � - " ' ' ' '

! � � ' ' - � � � & � ! � & - � & ' & - & � " & " &

! ! � ' ' ! - � " & ! & � & ' ' � 2 � & � &

' � � ! ! - " � " ! & ! - ! & ' � " ! � " &

' " � - ! & ' & ' � - " & � & � 2 � & � - � � " & "

! � ' ' - " & & � � 2 & " " & � ' & & ' 2 - � � & &

2 � � ! 2 & � " & & " 2 � & " & � & � ' & '

" & " & ' & ' 2 & � & " � 2 & ' & ' - ! " & " � - 2 '

' " 2 & � ' ' & & � � � " 2 & & � 2 � & & � " &

! ' " 2 � � 2 ! - - & � " " 2 2 - 2 � � ' - & � " - � 2 � 2 &

� � � ' " � " � - � � � & � � & " � & - 2 & � 2 & ' & & '

� � � " � � � - ! � 2 ! - & � 2 � 2 & & � � � ' - 2

" " � " � - 2 � - ' ' � ' � & ' ' " � " � ! - � - � & � � &

! " ' ' " � ! " ' � � � � � & ! - � � " - � " � �

- � ! " � � & 2 " & ' " ' & ' ' ' � � &

" � � & � � " � ' & - � & � � & � � & - & � - ' ' & 2 &

" � � & - & � - � " 2 � � � - � & � & " � " " � & ' '

" - ' " ' " � ' - & ! & - & & � " ' ' & & 2 - � 2 & & " � ' - &

� & - � & ' � � � � - " & & & 2 " ' - � " 2 & � &

� � & ! 2 - � 2 � - 2 ' & 2 & 2 - � & � � " & 2 - � ' &

� � - � ! & & � � � " & � - � - 2 - '

" � � & - & ' & � � " & - - 2 - � '

� " & " & � � ! & - 2 ! & � ' ' - 2 & 2 - - � 2

- " 2 � & � & �

- " � & - � � �

& � ! " ' ' � - & - � � " 2 - ' � - ' - � � � � �

' � 2 & - � � " 2 & " � & � ' � - " � & ' - 2 � &

& & " & " - - - � & � - " ' " & " - - � 2 - & 2

! - " � ! � - & - � " " ' 2 � " 2 � & ' & " ' ' & &

! � � � " 2 & " � & - ! " & � � - " " & � � � 2

� & � ' - ! & � ! - " � - � & 2 " � " ' ' ! - 2 ! & & � �

� ! � " � & " ! � ' ' " ' � 2 & � � & � � &

� � � � ! - � ! " ' ' � - & � � " - � & � � & - 2 " � � '

2 " � ' � - & - � � " � & � & � � � � � & ! & & � "

" - " 2 2 � ' - 2 - � � � " 2 � � &

" ' ! � & � " & & � � � " - � 2 � � � - 2 � " � 2 � �

! & " � - " � " � � & � � - " � - ' � & " � " & � �

� � � � - ' � & � ' � & � � - & � & � ! " - - " 2 � � ! & "

� - ' ' � � " - 2 - 2 & ! - & � ' � & � & � � ' & � " 2 � �

! ! � - " � � � � & � & 2 & & ' � � & � - 2 2 � � & ' 2 & �

" & � ' � � " & & ' � - - 2 � - 2 - & ' & ! & - � � � &

" � & - � � ' & & & & � & � " & " - - 2 & 2 " - &

Page 77: Goldrei.classic Set Theory for Guided Independent Study

� � � " � " * , . 0 * 0 � , * 9 , " , " � ,

@ � 9 9 " 9 " D @ , � " , 0

D @ � @ " 9 � 0 , 0 0 " � D 9 0 * 0 0 0 * 0

, 9 , " 9 " 0 0 0 0 0 " . " 0 D

0 " 0 0 * � " 9 D 0 0 * " .

0 � 0 D 0 . D 9 , " " , � 0 * � � " "

D 0 � " D D , � * , * , 0 " � D " D , .

" " 9 0 D 0 " � D 0 " 9 , � . � " * D * " * , 0 * 0 " 9 * D

9 , � . � . , D @ , " 9 0 0 � . " 9 " * D " 9 . . 9 , . " " * , * D

� * " , D 0 , " , D " * " 0 . � D 0 D

* @ , , D " @ 0 . . 0 " " 9 " . "

9 , " 0 0 0 " 0 * . 0 " 0 9

" 9 "

0 D , 9 " * " .

D 9 , D " . .

0 , " . " . .

, * D " " 0 " 0

" * D 9 0 0 " 0 @ " 9 * .

" 0 0 * � 9 " , 0 * " " 0 " 0 . 9 " " D

" , " D 0 � " " " . 0 * " . " � 0

D , " 0 9 0 " 0 . " D "

0 " " 0 9 0 . " "

" 9 � D . "

� 9 @ 0 * 9 " 0 9

D 0 D 9 " * " 0 * " 9 "

" 9 0 * , 9 , * " D 0 " 0 . . . " D 0 9 " "

, * � , 0 0 . " " * " * " . .

� . 0 D , 0 D " @ 9 , � * � , 0

@ , 9 , 0 � . " * 0 " " 9 * D 9 0 � 9 " " . 0 * 0 0 D

* 9 @ 0 " " .

* * " * 9 . " . 0

0 * " . " 9 0 * � " " 9 " " * 0

" 0 * . . 9 .

D 0 D . " . " 0 0

� 9 , " 0

* D 9 D 0 " 0 " 0 . "

, , D " " D " , , D � D D 0 0 D * D "

9 , , * @ , � � " 0 0 " 0 " . 0 * " " 9 D " 0 * 0 D

* , � * , 9 * . " " " 9 D " . � 9 0 * .

. . � 9 � " , � 9 , 9 0 * , * " , * * � . "

� D " * � , � , D

0 � " 9 * D " 9 . , , * " 0 " " "

* " 9 " 9 D 0 " , 9 " 0 . 0 " " " * " 9 �

D D * . " . " 9 0 0 * 9 . 0

Page 78: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � $ % � � � ( ( - � � ( $ � �

� � % % � - � � � (

- - � F F - � � F � ( F � � % � $ � � ( ( � � %

� � � % � % � - � F � - - $ � $ $ � F (

% � % $ % � � % � F ( ( ( � F

� $ � � $ % � � F % % � � ( F

$ � � � � F � $ � � � � % F ( F $ � % % � �

F � ( $ � $ � % ( � ( � F � F

� � � � F ( � % � � % � $ � % � � � (

� $ � � � $ - � � $ � ( - % � % � $ $ � � � � % � % � � � F � � F

( � � � � � � � � $ � % � $ -

� � � ( ( � � � ( � ( (

( � � ( ( � ( ( $

� � ( � ( ( �

� $ ( � �

( ( � � $ % % � � $ ( $ � � F

� $ - � % � � F � � � - ( $ ( % � ( � (

� � F � $ � ( (

� - ( � % % � % � � ( % ( F � �

� � % � � � F ( F F (

� ( $ ( $ � � � ( $ � F

% � $

( F ( -

( � F

� � � F

� � � � ( -

� � � F � $ � % F � � $ % ( �

� $ � $ F � % � - � ( � F ( � � (

� � � � � ( F % � F

� F $ � � F � $ �

� $

� � � � $ � � ( � - $ � $ � � � $ ( F % ( F � �

Page 79: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! # $ # ! � ) � ) � , ) ! ! ) $ ) / # � ) $ / ! , �

# / ) / ! � � � , � ) #

) / # $ # / # !

) / # # # ! ,

� , ) / � # � � � $ ) / $ ) � ) , ! ) / , , , / , � , ) , / ) � � ! # $

) / � � # ,

� � � # � � , / , ) , � � � # ! ! , � # , , ! , , � ) � � #

, ! � � # ! / , / � , , / , � $ ) � , � � ) ! ! � , � , ! ) $

# � � � # � ) # ) , ) , ! ) $ ) � , ) , ) � , # , , ! ) ,

� # � � � � / , ! # ! , � # � , , $ # ) , � / $ ) $ � � � , , $ �

# ) � , � ) � � , � ! � # � � # � ) , ,

� � # ! ! ) � , � , ) # ! ! $ # , � ) , / � # � ) / ) / # �

, / � # � # / � ! # / ) , / ) $ � # $ ,

# � � � # � � , � , # / # ! , ) / $ � � ) / ! # $ # # �

� # ! ! , / , # � $ , $ # ! � , # / # ! , ) / / $ # ) !

� / , � # ) $ ) � ) � , ! ) ) !

� � $ ) / $ / ) ) # � , , $ $ # # � # ! ! $ $

� $ # � / # � , $ ) , # ! ! / / # !

� � ! ! , / , � ) � � ) /

� # / , , / # ! , ) � � , ) ! ) / $ ) $ ! � �

, � � ! ! # ) / ) ! $ ) ) ) $ ! / � $ ) / ! � $

# ! , / , , $ ) $ , ) ) ! # $ ) ! $ $ ) $ ,

#

� # � ) , # � ) , ! ) $ ) / ! # � #

#

! �

/ , , / # ! , � # $

Page 80: Goldrei.classic Set Theory for Guided Independent Study

� � � � $ ( $ - � $ ( � 2 2 - 4 � � 4 6 $ $ 4 ( 2 2

� - $ 2 4 6 $ - - - ( 2 6 $ � � 2 4 � ( � 4 $ ( � 2 - (

( � k 2 k 2 ( � - ( 6 �

6 6 ( - � 4 $ 6 $ $ 4 ( � � $ ( $ - $ 6 $ 6 $ $ 4 - � � � $ 2

6 4 ( 6 � 6 ( � 2 - - - $ ( � ( k 2 � � ( 2 (

k 2 � - k � ( � � � ( k $ � $ 2 - � ( 6 � 4 $ � k 2 k k � 6 -

k - $ - $ ( � $ 4 $ � 4 ( � 4 $ � $ � � $

� 6 � k 2 k k � 6

� $ - - ( k $ � � k � 6 6 � 2 4 � � � 6 � 2 4 ( $ 6 6 � �

� k - $ ( � 4 ( 2 4 ( 6 $ - ( � 4 $ $ 6 $ 4

k 2 - � - $ ( 2 6 $ - ( � 4 $ 6 $ 4 (

- ( � ( 2 - $ � ( � 4 $ $ 6 $ $ 4

� � � 6 2 6 6 �

$ � 6 4 $ $ ( � � ( � ( $ � $ 2 - � 2 ( $ � � � ( 4

� $ 2 � - � � � k $ $ � 2 � � � ( k � 6 �

( � 6 $ 2 $ � � 2 ( $ - � $ ( 2 4 6 $ ( ( 2 - �

4 4 � 2 4 6 2 2 ( 6 4 � ( ( 2 �

- � 2 4 ( ( � 2 2 2 � 6 6 �

$ ( - � 6 � - $ � ( � � - � 2 4 � 2 2 $ � 6 6 $ ( $ � � 6

6 � - $ ( k $ $ � $ 6 � � $ 4 $ 6 2 � - � - � ( $ $ � - 2 6 6

- � $ ( k � ( � - - $ ( 4 2 � - $ ( $ ( � 4 ( � � 6 � -

� - � 2 2 � $ ( 2 4 ( $ 4 - 2 � 4 � 4 $ ( 4 2 � � 4 � 2 � � 2 $ $ 4 $

� 6 - 2 4 ( 2 � � � $ � ( $ 2 � 2 6 6 $ � k � �

( $ � $ 2 - � - $ ( � 6 6 - $ - $ � $ � (

k � 6 6 4 4 $ � 6 - � � � � � 2 � ( 2 $

� - $ � � 6 � 2 2 4 � � 6 6 4 � 2 � � 2

$ ( � � ( � $ ( ( 2 � 2 � 2

� $ � ( � � 6 $

2 - 6 � 4 � $ � $ 4 � 4 ( 2 4 - � - � � ( - � � $ ( 4 ( � 2

- ( � $ 6 � 2 ( ( k - � 6 � � � 4 � 6 2 ( � $ � ( 6 $ 4 2 - � 4

$ 6 $ $ ( �

( $ 2 - ( 4 $ � $ ( 4 - 2 2 4 � � 6 6 k $ - $ - 2 - $ � $ $ 4 ( $ (

� � 6 �

� 2 - ( � 4 $ - � � ( 2 ( � ( ( $ 2 ( 2 4 4 � $ ( � 2 4

- $ ( - ( $ � $ 2 � 4 2 4 ( $ ( � 4 ( $ � $ $ � � ( ( (

Page 81: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � % ( ( ( ( �

� � ( � \ \ � \

� � � ( (

� � \ � ( \ ( � �

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Page 82: Goldrei.classic Set Theory for Guided Independent Study

� � ! � $ % ! ( � � 4 � $ $ � � $ 4 $ $

4 A $ % $ � ! $ % � 4 $ 4

( ! 4 ! A $ A $ $ 4 $ 4 % A $ $ $

$ % � � $ ! $ � � A � % A $ $ $ $ $

( � $ $ % $ $ 4 $ �

( A A ! � % � � $ $ �

� $ %

! A A ! 4 ! A $ � % 4 $ $

! % � % A % $ � $ A � $ 4 � � �

! 4 ! $ $ � % � � A � $ % $ % $ % 4 $ ! �

! % $ ! � ! A $ � % $ 4 � $

$ A � % $ � � $ 4 $

! $ $ 4 % 4 � % � $ $ A $ $

! � � 4 � 4

! % � 4 $ $ �

$ % � % � % $ $ �

% �

4 4 A $ �

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� � $ � $ A

% ! 4 � % % � � � � $

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% $ � 4 $ � A � � $ $ � $ � A A $

% % � � A $ $ $ �

! % A A A A

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� | � 4 A

| 4 4 � � �

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� $ � ! ( � $ � � $ $ A �

$ $ % ( % % � % � $ � $ �

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( ! � ! 4 4 ! � $

Page 83: Goldrei.classic Set Theory for Guided Independent Study

L M N O R N W X Z L W O N f Z X i N Z N N W N N

N X O f L W f

L R Z f N R X Z N O L N N f O M N M N

O i N M N N

i N X L N N f Z W O N N f O R O Z f R L f L

N O N N f M N O N Z W X N X L X R W

O i M L f N O X L f X f N N R Z N O X W X X O W

N M L Z R f O X L L L L

Z L W W X L O X

N f N L X L R W L L

R X O W W N X L X W W

N W Z X f N N O X L W

N N N X X f X O O O R N O

f L N N N O M N N L L f R N f f N L

X L R N R O O X N Z N Z X N X f L f L O N W X

f N O X X W L L f L N W L L

i N N Z R N N O L O L L R X L Z R L X L

L Z X L O R L L

N N O X L N f O O X W L N W

L O O X M X M L O L

N X L L Z X X f X L W

N O M R L f X L L L X L N R W W L W

L M O O X f O R f M L L

W O L O i N X L L N O N X L f Z W X O N N O X L f Z W L X X f O N R O

X O N L M N N N L X X X X f N L W W L f N N R X N

L f N Z R X f O L N W Z O N L f N Z R L f X N N L X

O N X L i f N W R L Z O L X N R N L

N N f L f N O f L N X f L L N R O W Z

Z L X N L X L L

L W i N R N N X f O L R

Page 84: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � $ � � � � � � ( � � � � � � � � � � � . � �

� � � ( � � � � � � ( � . ( � � � � � � � � � . (

� � � � � � . � � � � $ � ( � � � � � � � � � ( � � � � � � � �

( � � � $ � ( � � � . � � � � � � � ( � � � . � � � � � $ � ( � � � � � �

� � � ( � �

� ( � � � ( � ( � � � � . � ( � � � � � � � � � $ � ( � � � . � � � � ( (

� � � � � . � � � � � � � $ � � � ( � . � � $ � � � ( � � � � � � � � � ( �

� � ( � ( � ( . � � � � � � � � ( � ( � � ( � � � � ( � �

� � � � $ � ( � � � � � � � � . � � � $ � � � ( � � � (

� ( � � � � � � � � ( ( ( � ( �

� � � ( � � � � .

� � � ( � � ( $ � � � � $ � ( � � . � � � � � � � � � � . � � �

( � � � � � $ � � � � � � . � � � � � . � ( � . � � �

� . � � � � � ( � � $ � � � � ( � � � � � ( � . ( � �

� � � � � � � � � � � � ( � � �

� � . � $ ( � � � � � .

( � $ � � � � ( . ( $ . $ � � � ( � � � ( �

� � � . . � � � . ( � � ( � � � � ( � ( � (

� � ( � � � ( $ � � $ ( � � . � � � $

� � � � ( ( � � � ( � � � � $ � � � � � ( � . �

� . � � $ � � � � � $ ( � � � � � � � �

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( � � � $ . � �

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� $ � � � � � � � ( � � � � � � � � . � � � � ( �

Page 85: Goldrei.classic Set Theory for Guided Independent Study

� � " % & " � ) ) � % � % ) " 7

< � " ) � )

) < 7 " " " � � � � ) � � %

" " 7 % ) h � ) � ) )

% & " � h " � � % " " � 7 7 ) � < < � � <

& " � % " � % % ) 7 " " <

� " < � " % � ) ) 7 %

� " % " 7 )

� " < " h ) h ) 7 �

� 7 7

) & h � ) 7 � � h

) ) ) �

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< " h � < % h ) h ) 7 � 7

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� h � " ) h � ) 7 ) � < ) � h % )

h 7 7 � % 7 ) ) 7 < �

) h � < % < < � < ) h h � h ) �

< � % � 7 7

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� " � ) % ) % � � % �

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7 " " < % ) 7 7 ) h ) )

< � ) " ) & " h < % < h " � h

h " � h " < % ) 7 )

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% � 7 % h " � � 7 7 7

� & 7 " % " < % ) � � � � ) % ) 7 �

h " � < % � � � h h % % � ) < ) ) �

Page 86: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � $ � � � ) $ � � � �

2 � � 6 6 � � � 6

6

) � � � � � � � �

) $ � $ � � ) � 6 � � � 6 � ) 6 � $ ) $ � � � � )

2 � � � � � � � � $ � � � ) � $ � $ ) $

) $ � � 6 $ � � ) $ �

) $ �

$ $ ) ) � 6 6 � � � ) � � � ) 6 �

� 2 � � � $ ) $ � � � ) 6 6 � ) �

� �

� � ) � 6 � � � � �

) $ $ � � )

$ 6

� $ � � 6 � �

� 6 6

� � ) ) � 6 � ) �

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� � ) � $ � � � 6 6 � � � � $ ) )

� � � ) � �

� � � ) � ) $ ) $ � � 6 6 )

� � � � � ) �

2 � � � � � ) � � $ � $

� � � � � � ) � 2 � ) 6 ) $

� � ) � � $ � � ) � � �

Page 87: Goldrei.classic Set Theory for Guided Independent Study

� � � � # � # # 8 : :

: � # # F # # 8 : � : : 8

_ 8

# 8 # � # � � # _ 8

� � _ # � F # � � � # 8 # # # � # : s :

: F # � # � # s : : � # : s # _ : # # _ F 8 � � # 8

_ 8 : � _ � : �

_ 8 : s � 8 _ F # _ � : 8 # :

# # F # # 8 � _ 8 � : � # F � :

� � � # : 8 F : # F # # 8 � 8 F

: � � � _ � � F _ s # 8 � #

� # � # s # � # s F � � � 8

s � s # # 8 � : s _ � s # s # 8 � _ � � : 8 : _ 8 : s # � � 8 _ s

F _ � � � # � _ # # 8 # 8 � # # � _ 8 � � _ � � � # _ 8 # # : # � 8

: 8 F : 8 # _ _ s # � _ F : _ 8 # : 8 s � : 8

# 8 F : � : s # 8 � � : 8 : 8 : s # s # _ � : # 8 # 8 : # s � � F #

8 : # # _ F # _ 8 : :

: � � _ � # � #

# # s � # :

� # � # : # # � � : : s � � # 8 # : 8 : : _ �

: s � � � # � # : s # s # � s � F # _ : _ 8

_ F # : � � � # 8 � � � : � # 8 � : 8 s � : 8 : # :

� F # : � � � : � 8 � � � # �

Page 88: Goldrei.classic Set Theory for Guided Independent Study

� � � � "

% ( ) � ( % , � ( 1 ) 1 , ,

( ) � ( % , K � � , � 1

1

1 % � � K 1 , � % , � � ) 1 , ) 1 � K , � ,

) ( % � � , " � " ) " " ) , 1 � , )

( % � � � 1 � " � � � 1 , , ) 1 ) � , � � 1 � � ) � K K , � �

� % � � � %

� , � K ( % K , � � � � ) 1 �

) K � K � � K , � � " ) � , � K

" � � ( K � , � � � � � � 1 " K K , , , � � )

� % 1 ) 1 " � K 1 ) ) ) � K , K ,

) ) K � K , " � , , ) ) , ,

� ( � 1 � � " � , ) 1 , � � � , , ) 1 � ) )

) ( � , " 1 � " , � � % , � , ) 1 ) ) K � K � K

% � , � ) � 1 ) � % � ) , K ) , � )

, % 1 ) � ) � � ) , ) , , , � ) 1

) K � K � � K , � ) )

K ) � � � � , � , 1 K , )

%

1 � � K � , ) " � , ) 1 � � , ) ) ) 1 ) K K �

" � � 1 � , " � , ) 1 � " % � � � ) % � � ( 1 � � � % ) " ) , � �

% ) 1 " � ) � ( � K � ( , 1 " % � � ) � ) ) 1 " � K , � � � " � , 1 � � ) ) � K � % , K "

� , � % K � , % 1 K � , 1 % K � � , % 1 ) " � � � � 1

( ) � K � � � � 1 ) 1 � � , , 1

Page 89: Goldrei.classic Set Theory for Guided Independent Study

J K L K N P Q S T P T K K Q

W K W L ] Q K W S N ] K K L L N Q T W Q P K K T T J

K K P L N P ] K

W L P Q K T T J K L K N S K W Q T N Q W P ] ] N Q J N J

J W ] S

W P Q S K T T J K N K W Q N T Q W P ] ] S K T S W

W L P Q K T J K K N P K T J N J T J K Q N W W P ] ] T K K ] K K Q T S

W

K W L K N S S N Q T J K S K S ] K T S N Q T L W W K N W P P Q P N ] N P L

Q W P N W Q S S W N P T K J T J K P J W J S T J K W L W L P ] ] S K T T J K L K

K T S J T J P T P P Q P L K Q T N Q J K N W W K K Q S W Q P N T

P ] W Q K ] N Q K S W K N K N Q T J K L K N W S S K T N W K P Q J W T J W L

P Q N K Q T K N S N K K P Q Q L W K W K Q T T N W L T J N

K K P K T

W K P J W L ] W S S K K L Q T W P K S K T K L N T

W L J K S K Q S J T P

K L N T K P ] ] K K S K T W W J K K T N Q S J J P T

K L T K P ] ] K K W W T J K S K N Q S J T J T

K K P S K T J W T J T J K L K S P K T J W S K K ] K K Q T S P K ] T K

S S K W

Page 90: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � ( � � � � � � � � � � � � � � � � � � � � � � � �

� ( ( � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � ( � � � � � � � � � � � � ( � � � � ( � � � � � � ( � � �

� � � � � � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � � �

( � � � � � � � � � � � � ( ( � � � � � � � � � � � ( � � � � � � � �

� � � � � � � � ( � � � � � � � � � � � � � � � � � � � � � ( ( � � � � � � � � � � �

� � � � � ( ( � � � � � � � � � � ( � ( ( � � � � � � � � � � � � �

� ( � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� ( ( � � � � � � � � � � � � ( ( � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � ( � � � � � � � � � � � � � � � � � � � � � (

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ( � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ( � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � ( ( � � � � ( � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � ( � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ( � � � �

� � � � � � � � � � � � ( � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � ( � � � � � �

� � � � � � � � � � � ( � � � � � � � � (

� � � � � � � � � � � � ( � � � � �

� � � � � ( � � � � � � � � � � � � ( � ( (

� � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ( � � � � � � � � � � � � � � � � � � � � � �

Page 91: Goldrei.classic Set Theory for Guided Independent Study

" # $ % % & % # " % ( " $ & $ & , $ $ , " / $ 4 " % 4 #

" # $ $ & % & ( , 4 , $ ( " & % & Q % ( 4 & / Q " , Q " # , $ $ , % " & $

& Q #

Q & , % ( # % ( " " , # " % % " & " % & , / / # / / & &

Q " " " % ( " $ , & $ $ , " / & / # $ # " /

( " & $ , & / , " " & " , & , " # & ( & , $ ( % ( " Q "

# " " Q 4 " % # & $ $ % " Q Q $ , $ & Q / , & $ % $ " " $

$ & ( , % ( 4 & % & , " $ , # , " ( # " Q " 4 $

% ( " , " ( % " " " $ , Q & # / / " & " %

$ , / / $ " " $ " &

4 " % , , , Q " , " /

" " " % % / " & " " $ % / & & , %

% " % ( #

" % " " 4 & Q , % (

( & ( % " Q " 4 # " , % 4 " / " " $

& % # % , # %

" " 4 / & / / & % & $ / Q " % & % , & $

& % " , , " $ %

& % # %

& # , ( " $

( & ( # % , # " % ( " $ & , "

" ( # " 4 & $ % & $ % Q , & &

% ( , Q " " % , & $ % , , $ % & # ( , 4 # #

% % ( , " & % " & " " $ , & $ , $ " ( # 4 $ " " $ # % ( & , " &

, & ( " & " Q % ( # ( " 4 4 % ( # " 4 $ " , , $ " " , " #

& # % ( " / $ " % # / 4 & $ " " ( " % & " 4 " # # % " & & &

" % $ 4 % / " " % % # , $ % & " , " # "

$ & & " " $ $ " # $ & Q Q " % 4 % " , & & " # # % , & $ % &

% % % , # " , $ % " " , % & $ " &

Page 92: Goldrei.classic Set Theory for Guided Independent Study

� � ! # ! % � # % ( - . # � ( 0 % . # � �

� � � # � ! � � 0 � % % � ! % ! . ( # 0 . 0 % � . % � - !

� - � 0 ( - � � ( # W � ( � 0 � # % . ( # � � ! - � � % ! ( - . ( . W �

� % - ! - # � � - � # � � - ! � - . % � - � % W � 0 ! - W 0

. ( ! - ! - W 0 � - � ( . # . ! ( 0 W ! % # W ! ! % W . � # � � ! 0 # ! �

. ! ! � 0 - 0 # . ! ( � � # # � 0 # � � 0 � # ! 0 � ! � . � # ! % �

0 # � ! � � % � 0 ( ! � ( ! ( % # - 0 % . # 0 W � . � # % . ( ! ( � ! -

! � ! 0 0 ( � � ( � # ! ! # 0 . ( # � � % � � - � 0 W 0 ( # �

� � ! ( � % � # � ! # # � � ! . # � ( 0 %

0 ( � ( � . ( ! - 0 W W

! - 0 W W

. % . ( � � % � # ( ( # % � ! # � 0 0 ( # � � ( % � # � � . !

! � 0 - 0 ! (

� # � 0 % � � ! � 0 0 � #

! # � 0 � � � � 0 W # . � ( . ! ( 0 ( . ( # � - % � # . ! ( W � # % � 0 W . # � # � � ! � � -

0 . W . 0 - ! � - 0 # ! ( ! ( % � # % ( 0 � W

( � . # W ! . W 0 # � # # ! � - . � # � � ! W � � ( # ! 0 % � # 0 % # � � � #

! 0 W W ! � # % ( ! . ( 0 � W % # . % 0 % � #

. % � # 0 ( . 0 W % ! % � # � � ( # � � . - ( ! ( ! W � 0 % � # #

# � ( . ! ( � W � � ( . � % � ! % � % % ! 0 # . 0 % � # � ! W � � (

% � # � . ! ! % 0 0 # ! ( # ! % � ! # 0 #

. % 0 � # # � � # � % # � W 0 � ! � � % � # . ( # � % # 0 ( ! 0 # ! - # �

! % 0 % ! ! - � # # 0 # % 0 % � # � 0 � � W % ! (

# � #

. 0 % � # # # � . � 0 ( # 0 # % % � W W 0 - 0 ! 0 ( � � - . � . (

� % . # � � ! W � � ( # . % 0 % � # . % . ( ! ( . % # � ( # ! - - � 0 % ! (

� � � % � 0 W W . % % W 0 � - # � . % ! � % ( ! # - ! � # � 0 # # � ! W � � # . % ( #

0 � # . # ! % � ! % % # � 0 # . # . % � . � W � - 0 W � # . % � ! W � # � �

Page 93: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � ! � � � � * � � 2 � �

! � * * ! � < � �

2 � � 2 � � � �

� � � � * ! � � � � � 2 � 2 � ! � � � �

� 2 � ! � � * � 2 2 � � * * �

� � � * � � � � 2 � � 2 * � � � � * * �

� � �

� 2 �

� � !

� * < � *

< � ! � 2 �

� � � �

� � � � * � � � � ! � �

� 2 � � * � ! < � � < < � < � �

! � 2 2 2 � < * 2 2

� * ! � � * � � 2 * * � � � �

! � � �

� � * � 2 ! � �

� � � � � * � �

� ! � � � � � < * � 2 * � 2 * 2 � � �

� � � � 2 � � � � � � �

2

2 � �

� � *

� � < < * �

< ! � � 2 � 2 ! �

Page 94: Goldrei.classic Set Theory for Guided Independent Study

� � � � � ' � * � . � / 4 6 � � � 6 * � � ' . 6 9

* 4 � . 6 ' � � � * � � / . ' . ' � 9 � * � * . � ]

4 � � � 6 9 * � * � ' � 9 � 6 * � � ' . 6 9 * 4 � ' ' � � . � / . 9 � 6 9 6 ] ' 4

* � * � � 9 � . 4 * 6 9 � � � ] . � * � ] 6 � 4 9 ' � 4 * � * ' 9 � . 4 9 ' ' � � '

4 * . 9 ' � 6 9 � / � 4 * � * 4 9 6 9 � * � * � ' � 9 � 6 ' 6 ' � �

. . ] ] � ' � � � � . 4 4 ' � 6 6 � . 4 9 4 9 � � 6 � ' 4 9 6 �

. ' � � 4 * � � � � 4 � / . � � . 4 9 � ' 6 ] ] 4 � 9 � * 6 � � 4 9 4 6 * � � ' . 9

* 4 ' 6 * � � � 4 * � * � � 9 � 9 � 4 9 ' 6 ' ' � ' / � � 4 ' � * 6 � . 4 9 4

� � 6 � / � 6 � . . ' � / . * 6 ] 6 ] ] � / � . 9 4 * 6 � . 4 9 4 � 6 9 . 4 9

' 4 9 � 6 . 9 � . 9 � / � ' � 4 4 * � * � 6 . ' . '

� � ' 4 � 9 ] ] � � � 4 � / � 9 � . 4 9

. � 9 6 ' ' � � 4 4 . / � � � / � � / � * . � . ' / 4

� 9 . 4 . / 4 6 . 9 6 9 4 4 6 . 9

4 � � 6 9 � . 4 9 . � 4 6 . 9 � � . * � � � 4 * � 6 � � . '

9 . � � ] � � 9 � 4 / / 6 ' 9 � * 9 � � 4 4 * � * � 6 . * ' . � / �

* 6 4 � . ' � 9 ' 6 � � / � * � . ' 6 9 . � 9 6 � ] '

� / 6 � ' . � / � * 6 / 4 . � 9 6 ' ' � � 4 . ' � * 6 / 4

' 4 � � . 4 9 . / 4 6 . 9 . 4 � / � � . ' 6 9 . � '

� / 6 �

9 � � ' 4 4 4 * 6 ] ] 6 9 6 � . 9 . / � 4 ] 4 4 � ' 4 * � * �

' � 9 � 6 � . 4 9 4 * � * � 6 * ' ' � � / . 4 9 . � . 4 9 4 9 4

� / � ' � 4 9 � 4 ' � * ' � 4 6 4 ] � 4 � 9 . � . 4 4 9 . 4 9 '

* � * � ' � 9 � � ' � � ' . � / .

� � 6 9 � ' � ' . � ' 6 � 6 � . ' 6

.

Page 95: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � " � � # $ � � $ ) � # � " � " ) $ " $ � # 2 � $ � � 2 " $ # $ 6 6

� ) � " � 6 # ) � 6 � # ) # � 6 ) # � " � # ) � $ " � � # ) � $ # 2 )

� � � " " � � $ 6 " � $ $ � � � � 2 � # $ " � $ � $ " 2 � � � ) � � # 2 2

� " � 6 � � $ � � $ " # � ) � 2 � � � $ " � � 2 � #

" � � ) � � " $ 6 � � 6 � � " 2 � 6 6 # � � � � # 2 � ) � $ � ) # )

� # � � # ) # ) " � " ) � $ 2 # � � � $ ) � $ # 2 � ) 6 6 �

# ) � $ # � # 2 $ � # � " � 2 � � # 2 � � 6 � ) 6 � $ #

2 � $ ) ) " � � � � # $ ) 2 � � � � # ) " � $ � " � # $ � $ 2 2 � 6 $ � � )

2 2 � � � # 2 � � � � � � # � ) # 2 � � � $ � � " �

$ � � � " � ) ) � � � $ � " � � � $ � $ 2 $ 2 � 6 6 # 2 � � �

" 6 $

$ � 6 � 6 � � � # ) $ � � # 2 " # 2 � � # ) � 6 6

# 2 � " � � " ) � ) � $ # � � ) � 6 6 " � � "

) � 2 2 $ " � � # $ " � 6 " ) � � � � ) $ )

� � � # ) " 6 � # � � � $ � 6 6 6 ) " # ) � � � � � 6 # � ) � $ )

� � # $ ) � $ � � � # $ ) � # � 6 " ) $ " � � � ) $ � � " � #

� � � � # ) ) � � � � � $ � " " � # ) � � $

� ) # $ ) � � " ) " � # � � �

)

� � � � � " ) � � # 2 � � ) � # � � � # � " ) $ � # � # 2 � # $ ) � " " $ � 2 �

) 6 6 ) � � � $ # ) �

) � 2 � # � " � $ " � � $ ) �

2 � ) � � $ ) � � # � � � " � � � $ " # � # � � �

� � �

) " � 2 # � � # ) ) � � ) 2 �

� # $ ) � # � " # � � #

� � � � � 6 $ $ � � ) � # 2 # 2 � � " " $ ) � �

) � " " � $ ) � �

) # � # � $ 2 # $ � � $ # $ ) � ) � � $ ) � " � " � # ) � $ " $ � " � $ 6

� ) � � # # ) � # $ � � � � � 6 � # 6 2 � # 2 � � � � � 6 " " � # "

Page 96: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � " # 0 � # � � 5 � � #

# # ? � ?

#

? �

� � � s � ? � " #

� � # " s " � � ? � 0 0 � 0 � 0 � 0 " 0 �

? � " # � � 0 # � 0 �

" # s 5 � 0 � � � � � s ? � " #

� � � ? � ? � # # # � ? � s " � ?

" " � � # ? "

� � s � # # ? s # # 0 # 0 �

# 0 ? #

# � ? � " � " # ? � s 0 " � " � s # " " � 0 s # � # " s

# � � � # s "

" # 0 � ? � � # � " # " s � � # ? 5 " # 0 # ? #

s � � � 0 s � � " # " s � � # # " s � 5 � � #

� " 5 5 s � # � � ? ? � � s " � " # # s � ? 0 ? � " # ? # 5 5 #

� � � s � � � � � � s � � # ? 5 � # 0 # � # " � � ? s

" s � �

" s ? � " #

� � � # ? � # � ? � � # 5 # # 5 " � � # � � " ? � " # � " � " � � ? # " 5 5

# � 5 � ? 0 � 0 " � � � 0 � � 5 # # ? � ? 0 " ? 5 ? 0 " � � " # 0 � � �

� � 0 � � � # � � ? � � � ? � � � 0 # � " # s 0 # ? � 5 � � �

# ? � � � � � � � � " s ? � � s � # � ? � #

# � � � s � � ? 5 5 � � � # s

� � � # s 5 � � ? # 5 0 � 0 � s � � ? # # � � � � � s

s � � � ? � � ? � ? ? s � � # # 0 � 0 � ? " 0 � � "

Page 97: Goldrei.classic Set Theory for Guided Independent Study

� � � ! & � � ! � � ! � < <

> @ > @ � < < � � &

� � &

� > � & � �

< < ® � � � < < < < � � � ®

� Ä < @ � Ä � �

> @ ® @ è & @ � & � ® � < � Ä � � & & Ä è

& � Ä ! Ä � ® � & �

� è ! Ä & Ä � è � & Ä ! @ � ® � � Ä

Ä @ è Ä @ & @ � � & & � @ � � �

@ è ! < < < � � @ & � � & < � & � � �

@ ® @ < � ! � & � � & � � � < �

< ® � � & ® � � � ® � � � �

® � � < �

� Ä � �

� & � Ä � è < � < < �

@ ! @ ! � � � @ � � & Ä & @

� > � è � � < ! � � � � & � & � < < < ® ®

� & ® � � < �

< < ® � <

& � Ä

� ® è Ä Ä ! � & � & � � � �

Ä > Ä � � & � ! � � � <

& � < � & � � ! < < < è � � ! � Ä � & <

! ® ! � > ! Ä � Ä > ® � � � � Ä Ä �

& @ < @ @ & Ä � ® @ � Ä Ä < < � �

� � < � & � & < ! � � &

� & Ä ® Ä � � & < � è ® Ä � � � �

! � & @ � & �

� � Ä � �

� � ® � @ < � Ä Ä � �

� & � Ä Ä � � & � ®

! � � è � @ < � � � Ä � � & < � �

< ! è � < Ä @ ! � � � Ä @ � � Ä � <

� � ® ! & Ä ! Ä � � � è Ä Ä � � ! � � ® �

Ä @ ! Ä � @ � ! < < Ä @ Ä � ® � @ è � Ä & Ä ! < & � Ä ® !

� � & & � Ä @ è @ Ä è & Ä � ! � � & @ > ! ® � � Ä ® � Ä � � � Ä

Page 98: Goldrei.classic Set Theory for Guided Independent Study

4 5 5

M N O 5 M T V W 4 4 M N T M 5 N O 5 4 T

5 O V O N N T O V M W N 4 N O W 4 W 4 5 W T T W 4 4 T N

� V M V 5 5 V O W O W N 4 T N

5 4 M 4 V M W � M O V T 5 M T 5 M N T O N M 5 T

M ¾ M

4 W T T 4 N T 5 5

M V M

¾ T ¾ 4

O

M O N T N W 4

4 O W

N W ¾ T 5 4 5

4 N W M 4 4 N W

4 4 5 5 5 M

5 5 N 4 5 T

4 T T ¾ N 5

T T T

W W

M O W 4 N O 5 N V 4 4 N M 4 V T T N 5 N

W M ¾ O W 4 N 5 4 N N 4 5 N 5 V M T O 4 T N ¾

� T O W 4

4 N 4 4 5 4 T 4 T T 5 N W

4 5 M N M 5 4 T 5

Page 99: Goldrei.classic Set Theory for Guided Independent Study

# $ ' $ * # , ' . ' * ' ' * ' ' < # , # ' * # ' ?

? ' , < ? $ . # . , . $ # *

$ , . $ . , < ? # * ' <

$ < ' � # � # �

< # * , ' < ' � $ * # # # < # * ' # < , # � # < * � # � # , . $ , # , ' <

' � ' # � ' � $ , . < , * # , . � # # < * ? #

' � $ , * � , < ' � ' � $ ? ? $ < #

# ' ' # * ' * # ? * # # , ' . ' # . $ �

$ * # � . < # , ? ? # < . � # ' * * * . $ $ * � # , < , #

* � # ? $ * # � # , < * # $ , < ' , ' * # ' ' # ' * � ' $ ' < , � $

' � # # < , $ ? * ' < # ' * # , ? ? � $ # � # # < * . < ? , < < # � , <

< , < # , < * , ' < $ < # . � # � . , # ? , * , < , ? * ' * # � , ? ?

, # < $ # # ? # . * ' # , ' . # $ ? ? * # #

, ' $ < # ? $ , < # , < # $ , ? ? $ # � ' <

� # � # , . $ < , < * # . # *

� #

' � $ ' . , ? � # # � � , < * ' < $ . # * . .

, < , ' * � < # < , ' < ,

� # . , * # * ' # # . ? * , < , # '

# � # * , . , # ' < * $ , < . $ < , < , ? # ? # # <

� $ ' # . # , ' ' , < < , $ # * ' ' , * , < < , * , ' �

$ < * # # . * $ * � # � # . , < * , # # . ' * $ # , < *

$ < ' * $ * $ # , , < # $ < # < # ' , #

, < < , * # , * , , < * ' * � , . . # * . ' < , < * , ' < < # <

* # < . � ' * $ * * � # # * , * # ? , , < , * # $ , < # , ' , ' < '

Page 100: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� > � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � > � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � > � � � � � � � � � � � � � � � � � > > � � � � � > � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � > � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � > � � �

� � � � � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � > � � � � � � � � � � � � > � � � � � � > � � �

� � � � > � � � � � � � � � � � � � � � � � � > � � � � � �

� �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � > � �

� � � � � � � � � � � � � � > � � � � > � � � � � � � � � � � � > �

� � � � � � � � � > � � > � � � � � � � � � � � � � � � � �

� � � � � � � � � � � > > � > � � � � � � � � � � � � � � � � � � � � � � �

� � � � > � � � � � � � � � � � � � � � � � � � � > � � � > > � � � �

� �

� � > � � � � � � � � � � > � � � � � � � � � � � �

� � > � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � > � � � � � � � � > � � � � � � �

� � � � � � � > > � � � � � � � � � � �

� � � � � � � � � � � � > � � � � � � � � � � �

� � > � � �

� � � > � � � � � � � � > � � � � � � � � � � � � �

� � � � � � � > � � � � � � � � � � � � � � � � � � � � � � � � � � � >

� � � � � � � � � � � � � � � � � � � � � � � > � � � � � � � �

Page 101: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � ' � � ' 2 � ' ' � 2 � � � � 9

� � � � � � � � 9 � ' � ' � � � � � � � � � � � �

2 � 9 � � ' 9 � ' � � � � � � � 9 ' ' � � � 9

� 9 ' 2 �

� � 2 ' ' � � 2 � 9 � � 9 � 9 � � ' � 9 � � � 9 ' � � � 2 � � 9 �

� 9 � � � � 9 � � � � � 9 � � � � � � ' � � � � � 9 � � � � � � 2 2

� � � � � � � � � � � ' 9

' 2 � � ' 9 � 9 ' ' � � � � � � � � � � 2 ' 9 � � � � � � ' '

� 2 � � 9 � � � 2 2 � � � � � � � � � � � � � � � ' 9 � � � ' � � � � � � � �

� � � � � � ' ' � 2 � 9 � ' ' � � � � � � 9 � ' 9

� � ' � � 9 � � � 9 � ' � 9 � � 9 � � 9 � ' 9 �

� � � � � ' ' 2 � � 9 � � � � ' 9 � � � 9 ' 2 2 �

' 2 � ' � � 2 2 � � ' � ' � 9 ' � �

� � � 2 � � 9 � � ' � � � � � ' � � � � � ' ' � � � � ' 9

� � � � � � � � � � 2 9 � � ' �

� � � � 9 � � � 9 � � ' 9 � � � 9

� 9 � 2 � � � ' 9 ' � � � � � � � � � 2 2 ' � � � 9 � 2 � � 9 � � � � � 9 � '

� � � ' � ' � � 9 � ' 2 � 9 � � � � � � ' '

� � � � � � � � ' � � � ' 2 �

� � � � � 2 � 9 � � ' � 9 � � � ' 9 � '

� � � � � 9 � 2 � ' ' � � � ' ' � 2 � �

� � � � � � 9 � � ' 2 2 � ' � ' 9 � 9 � � �

� � � �

� � � � 2 2 2 ' ' � � � � ' � 2 � � 9 ' � � ' ' � 2 � � 9 2 � � � 9

� � � � � � ' 9 � � � � 2 � � 2 ' 2 � � 9 � � 9 9 � ' 9 � � � � 9

� � 9 � � 9 � � 2 � � � � � � � ' 2 � ' � 2 � � � 9 � � 2 ' � 9 � �

� � � � � � � 2 2 � 9 � ' 9 ' � � � � 2 2 � � � � � � ' � � ' � �

� � 2 9 � ' � � � 2 � � 9 � � � 2 ' ' � � ' � � � 9 ' 9 2

� � � ' � 9 � 9 � � ' 9 ' ' 9 � � � � 2 � ' � 9 � � 9 � 9 � ' 9 ' � '

� � � 2 � � ' 2 � � 9 � � � ' 2 � � � ' �

� ' � 2 � � ' 2 � 2 2 � � ' 2 �

� � � � � 9 ' � ' � � 9 � �

� � � � ' � � � � � 2 � � � � � 9 � 2 � � �

� 9 � � � � � ' � � � 9 � ' ' � � � � � �

� � 9 � 2 � � 9 � � ' 9 � � 9 � � ' � ' � � � � � � 9 ' � � 9

� � 2 2 � ' ' 2 � ' 9 � � � 2 � � � 9 � ' � � � � ' � �

� � � � � � � � ' � � � � � � � � � � 9

� � � � ' 2 � � 9 ' � 2 � � ' 9 � 9 � � � � � 2 � � � 9 � � ' 9 � � � ' � 9

� ' � � � � � ' ' � 2 � � 9 � � � � � � � � � �

Page 102: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � ! � � � � � � ! � � ! � � � �

� � � � � � � ! � � � � � � � � � ! ! �

� � � ! � � � � � � � � � ! N � � � O � � � � � � �

� � � � � � � � � ! � � � ! � ! � � �

� � ! � � � � � � � � � ! � � � ! � ! � �

! � � � � � � � ! O � � � � � � � � � ! � � � � �

! � ! � � � � � � � ! � � �

� � � ! � ! � ! � � � � ! � ! � � ! N � � O ! O

� � � � � � � � � � ! ! O � � �

� � � � � � � ! � ! � � � � ! � ! �

N �

� � N � O � � � N � � ! � O � � � � � � � ! � � � � ! � � �

� � �

� ! � � � � ! ! � � � � ! � � O ! � � �

� � � � � � � � � � � ! ! � �

� � � � � � �

! � � � � � � � � � ! � � � � � �

! � � ! ! � � � ! � ! � ! O

� � � � � ! ! � � � � � � � � � � � �

� � O � � � � � � � � �

� O � � � � � ! � ! � � � � � � � � � � � �

� � � �

� � � � � � � � � �

! � � � � � � � � �

� N � � � � � � � � � � � ! � � � � � � ! � �

� � � O ! � � � � O � ! ! � � � � ! � � � � � � ! � � � ! � �

� � � � � ! � O ! �

� � � � � � ! � � � � � �

� � � � � �

� � � O ! � ! � � �

Page 103: Goldrei.classic Set Theory for Guided Independent Study

# $ & $ ' ( $ ' & 1 $ ' 5 ' 7 8

< $ $ 7 # 8 ' # $ 1 < 7 # 8 5 1 $ ( 8 < 1 7 8 1 1 '

1 & < 1 7 $ 8 7 8 $ ( 7 $

& $ ( $ < ( $ 8 ' # $ 1 # 1 & < < 8

1 $ 8 1 $ ( $ 7 # 8 7 8 8 7 1 $ $ # $ 8 7 8

' 8 1 ( 7 7 8 $ ( 1

7 # ' 8 $ 7 1 ( '

< $ < 8 # $ 1 < 7 $ 7 8 5 $ 5 $

# ' 8 7 # 7 5 5 7 ( 7 1 ' 5 1 ' 8 8

< ' 8 # $ $ 8 $ ' $ ' ( $ < 5 5 # < 7 8 $ 8 7 1 $ 5 ' 8

# $ 1 < 8 & $ ' 1 1 ' $ < < 7 8 1 ' < $ 7 1 '

$ 5 $ $ 1 ' < 8 ' 8 $ & $ 1 $

' ( ' 8 $ ' 1 $ # # 7 8 < 7 8

& < 1 # ' ( ' $ 1 7 1 ' ' '

8

1 7 # 1 $ 1 7 8 1 < 1 1 ' $ 1 7 1 7 ' ' $

' ( $ ' 5 7 < 1 $ 1 8 7 8 8 $ 5 7 1 < $ 5 ' ' $

< 7 8 # ' 7 ' 8 ( 7 1 8 ' # 1 1 $ ' 8 $ $ $ 8

8 # 7 1 $ ' 1 & 7 # ' 5 < 1 7 8 ' 8 $ $ 8 $ ' 1 $ 7 ' ' 1 7 ' < 5 '

7 1 & 1 & $ ' 1 & $ ( 7 ' ' 7 < 1 5 5 $ 1 $ 7 5 ( ' 1 1

# $ 1 $ $ 1 & < # 1 ' 1 7 # 7 ' ' 8 $ 8 ' 1 1 $ 8 7 # ' $ 1 #

$ ' 8 # 1 ( 1 $ 8 7 1 $ ( < 1 7 ' # # # 1 < 1 7 8 ' $ $ 1 1 8

$ $ < 1 $ 5 ' ( 7 8 ' $ ( # 1 < 8 8 7 ' 8 7 # ( ' ( 7 8

Page 104: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � - � 0 � � 0 �

0 V

p - 0 � � � � 0 � � � V p � � � � � � � � 0 V � -

� � � � p � � V � p � � � p � � p 0 - � � � - � 0 � � p � 0 0 V � � p � � - � � � �

p � � � � � � p � � � p - p V V 0 - � � � � V 0 � � � � � p � � � � p � -

0 � � � � � -

� p � - � � - � 0 - 0 � � p � 0 - � � � p � � � � � � p V � � �

� � - � - 0 � V � p � - p - � 0 - - � � 0 - � p � � � V � � � � � �

� � - p � � p � V � p - � � � � - � 0 0 - � 0 � � � � � � � - � � � � � V p V 0 �

� � V � � � � � � 0 � � p p 0 � p � � � � - � � � � � � � � � � p - � p

0 � � 0 0 - 0 � � � � 0 � p � � p - 0 - � � V � � � � � � � � - � � � � V p V �

� � � � � � � � � 0 0 � � - 0 V � - � � � � � � � p � � � � � � � - p � V

� 0 � � � � � - � - - � � 0 - � � � - � � 0 0 V � 0 � � 0 - 0

� V � � � � p V � � - � � � � � � � � � p - � � p � - � - 0 - 0 � � � �

p � - � � � - - � � � � 0 � V � � � - � � � 0 V �

0 � - � � � � p � � V � � � p � - 0 � � p V 0 0 0 p � � - 0 � � � V � V 0 -

V � - � � 0 � � � - � � V � � � p � � � - p V 0 0 0 � � � � �

� � p � � p � � � p � - � - � � � � � � � - V � V � - - 0 � � p � � � � �

p � 0 0 � - - p - 0 � � V 0 V p - � � - � � - p � - � 0 -

� V � � � - � � � � � � - � � � � � � p � 0 � - � � 0 - p V � �

� V V � - p � � � � � � � p V � 0 p - 0 - � V � p � � � � � 0 �

� 0 0 � � � � - �

� � � p - � � � � - � 0 p � � � p - p - 0 - 0 -

p � V � � - � � V � � � � p - � � - � � 0 -

� � � p �

0 V p � �

� � � � � � � � � 0 V � � V 0 � � � 0 � � � V 0 0 � 0 � p - p � � 0 p

- � 0 - � � p V � p � 0 � � � V � � - � 0 � 0 � - � - � �

Page 105: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � ) * . / 0 1 ) � � ) . � 0 ) �

. . : � / . ) �

. . / 0 1 ) � � . 0 1 1 /

. .

. ) � � � . : : 1 � 1

1 � � � : 0 ) / � / / 1 * 1 . � � 1 � / . ) � � 1 / : : ) * 1 � 0

� � / / 1 : 1 � � 1 * � 1 . � 0 � .

1 : � � � / 0 1 � 1 � 1 / 1 . ) ) / . : � . � / 0 . . : 0 1 / 1 / / * 1 . * 1 :

0 1 . ) ) * 1 � 1 1 � � � 1 0 � / . / : / 1 1 * 1 * ) 1 * . 1 . * :

) � 1 � 1 1 / 1 � ) . � � 1 �

) : 1 / 0 � . � . ) � � � � � � / . � /

1 � � / 1 �

1 : � � � : 1 . � . ) � : ) / / ) � �

1 � 1 1 � / ) ) / 0 � � � * 1 1 � 1 / ) 1 ) / 0 1 * 1 � /

0 ) � : * ) : 1 � ) 0 � / 0 1 * 1 � / 0 ) � : ) * ) / 0 � /

1 0 � : 0 ) 0 � / . / � � : ) ) � : * 1 0 � 1

� . � � � . ) �

) * : / . ) � � /

: � � � . ) �

0 : ) : / � / 1 1 : � / ) � ) � 1 . * 1 � / . ) � . 1 /

/ 1 * 1 � / ) � ) * � � � ) � 1 / . � 1 ) ) ) / 1 � 1

) � : 1 1 � 1 ) . � 1 � � � 1 � . ) : ) /

. � / 0 1 � 1 � 1 � 0 . / 0 1 * . � � ) � . /

� 1

1 � � � � ) : ) / � / 0 1 * 1 . : � � � / . ) � . 1 : 0 � � � � ) 1

1 0 � � ) / . . � / ) � ) �

) * 1 . * � / . ) . / ) � . / 1 ) � � 1 1 � / )

/ 0 1 ) � . . � � � 1 1 . ) 1 / � / ) ) � * � / 1 / /

. � / . ) � ) 1 � 1 � 1 ) ) 1 1 � . *

. . � 1 1 � � / � . / 0 ) � . � � / . 1 . * 1 � / : . . . � �

. / 1 � � / / 1 * � � ) * � / . ) �

) * / 0 1 . � / . 1 / 1 1 / 0 � � � / . ) � 1 . : ) � � : 0 )

/ / 0 1 1 . : � � . � � � 1 � 1 1 / ) ) . : ) � � � 1 / * ) * 1 *

� . * ) ) / ) ) 1 / 0 1 1 / * 1 � 1 1 . � / 0 1 ) . � �

� / ) � / 1 1 / . . � � ) 1 1 * 1 . 1 1 / . ) � �

� . ) 1 1 �

Page 106: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � 4 � � � 4 4 � � 4 � � � � � � � 4 � � � 4

� � � 4 � � � � � � 4 � � � � � � 4 � � � 4

� � � � � 4 � � 4 � � � � 4 � � � � � � � � � � �

� � � � � � � � � � � � � � 4

� � �

� � 4 � � � � � � � � � � �

� � 4 � � � 4 � � � � � 4 �

4 � � � � � � � � � � � 4 � � � �

� � � � � � � �

� � �

� � � � � � � �

� � �

� � � � 4 � � � � 4 � � � 4 � 4

� � � � � � � � 4 � � 4 4 � � � � � � 4 �

� � � � � � � � � � � � 4 � � � � � �

� 4 � � � � �

� � � � � � � � � � � � � � � � 4 � � � � �

� 4 � � � � � � � � � � � �

� 4 �

� 4 � 4

� � 4 4 � � 4

� � � � 4 � � � � � �

� � � � � � 4 � 4 � � � � � � � � � � � 4 � �

� � � � � � � 4 � � � � � � � � � �

� � � �

Page 107: Goldrei.classic Set Theory for Guided Independent Study

$ % $ ' $ $ / / 0 0 3 % ' $ 0 3 8

; ' ' ; % 0 8 ; 3 % 3 % ' $ 3 $ 3 $

0 8 ; 3 3 0

' ' % % 3 % 0 3 ; 3 ' $ % ' $ $ 8 ` $ ` $ ` % $ 8 ' $ ` $

% ' $

3 % ' $ ; ; 0 3 % ' 8 $ ` 8 0 3 % ' $ ` $ ` 8 0 3 ` 0 3 0 / $ 0 8 3 3 $ `

3 ' $ ` ' $ ` $ $ 0 3 ' 0 ' 8 ' 8 ' ; % $ ; 3

8 $ ` $ 8 ` ' 0

` 3 % ' 0 ' ; $ ` $ / / 0 $ % 0 3 ` $ 3 $ ` % ' $ ` $ $ 3 ` $

$ / $ ` 3 % ' $ ` $ ` 8 0 3 ` 0 3 0 / $ ; % ' $ ` % ' ; 3 8 % ; % $ 0 % ` ; 0 ' % ; ;

/ $ % / ; % ; 8 ; % 0 3 ' 0 % 0 / / ` $ / $ $ $ 3 % 0 3 $ $ ; ` / 0 $ ` % ' ; % % ' $

` $ / ; $ $ 3 % ; 8 ; $ % $ ` $ / 8 8 ' $ 3 0 % 8 3 % ; %

% ' $ / 8

/ % $ 8 ' $ ; 8 $ % 0 8 / ; 8 8 ' 8 ; ` % 0 / ; ` 8 3 ; 3 $ 0 % 0 3

; 3 ` 8 ' $ 0 3 3 0 % $ 8 $ % 8 ' $ $ % $ 3 % $ ` $ % 0 3 % $ 0 3 8 $ % $ /

/ $ ; ` $ ; 8 / / 8 8 $ ' $ ` $ % $ 3 $

% $ 3 $ ; 3 % 0 3 0 % 0 3

` / /

' $ 3 ; 3 $ ' 0 ' 8 ` $ 0 8 $ / % ' $ / ; 8 8 $ ; 3 0 ; $ %

' $ ` $ 8 ; 0 3 ` ; ` $ 3 % $ 3 0 3 % ' $ ; $ ' ; % 0 8 0 %

' $ 8 ; % $ $ 3 ' $ ` $ ` $ 0 ` $ 8 ; $ ` ' 0 0 ;

3 3 % ' 0 8 ; 8 $ / ' $ % 3 / $ 8 ; $ % ` $ 0 8 $ / % ' $ / ; 8 8

' 0 ' $ ; ` $ % ` 0 8 ' 8 ;

8 $ %

3 $ 0 ' % $ 8 0 / % 3 % ; % ' 0 ; $ ; 3 8 % ' % ' % % ' $ ` 0 8 $ / 8 / $ ;

$ 8 0 ` ; / $ ; 3 3 ; % ` ; / ; % $ ; 0 ; / $ 3 0 % 0 3 0 $ 0 3 ; % % $

% ; $ $ 3 % $ ` $ ; 3 $ $ ` $ / ; 0 3 % $ 3 % 0 3 ; %

$ ' ; $ $ 8 ` $ $ ; / 0 $ 8 / 8 8 3 % 3 ` $ 8 3 3

% ` / 8 ' % ' ; % ` $ ' ; 0 ` 8 $ 8 % ' $ ` $ 0 ; 3 0 $ $ %

8 ' % ' % ' / 8 $ ` $ $ / % ; $ % $ % ' $ / ;

' 0 0 $ / 8 % ' $ / 8 8 3 % 0 3 ' $ ` $ ` ; / / ; / /

8 $ 8 ' $ 0 $ ` % ' $ % ' $ ` $ 0 8 ; 8 / 8

Page 108: Goldrei.classic Set Theory for Guided Independent Study

- - 8 > 8 ? @ - A A B - ? C @ B A 8 A ? C - B K

@ - A B - C - N @ K N - @ - @ A B A B ? > @ @

- - K A > - - > 8 K ? K

- C - - - C @ - ? @ K > - A 8 N - K B - K A C -

N @ 8 K N - N K A 8 N K @ B A A

? C 8 > >

B - C ? ? N @ - 8 A > B - @ 8 - 8 @ A B 8 A ? B - ? C - B - ? K > > 8 - B - C -

? K - B A ? - @ > N B > ? C - ? K @ N ? @ ? 8 @ K ? K N @ A B - @ A 8 - > - A @

8 ? - C @ - N K @ B N K 8 A - K -

N 8 A > > 8 @ - B N N C A ? B 8 N C - N - ? C N A ? - 8 K A N ? K

@ B - N K ? K B ? B - @ N @ 8 A N @ A 8 - ? A - C ? ? 8 @ B 8 N @ 8 K

A N - 8 K @ A B A B - - N K N - - A K 8 - > @ A

? > - K B - ? - C - 8 N C ? K @ A C N ? K - 8 - A B 8

N @ - @ A B A B - @ K > - A ? K N @ - A

N K A A A B - K N K N @ @ -

- C - - K A > @ N - B - @ - ? A B A - ? C - @ 8 N K A 8 - C - - K N K 8

@ - - K - ? @ - A @ C 8 A B - C B 8 K @ 8 N K B A - - ? K @ C A N K 8 K A N ? K

N A B 8 N K N - N 8 A - ? K @ - - K - ? - A - ? C - N A B 8 A B -

B 8 @ K N 8 - - B N - ? > C N - N K A - C @ ? A - 8 @

B N @ ? C A ? ? K @ A C N ? K N @ ? ? ? K A B A - @ > > B K ? C N A @ A - ? > > ? N K

> ? ? - > @ - A B - C - N B N @ 8 @ N > - ? K - - K - ? B - ? C -

? @ - 8 N @ > K A N ? K 8 K > - - 8 N - K @ - A

B - K @ - - K - ? @ - @ 8 K - - K -

? C 8 >

@ 8 @ - A

- B 8 - ? @ - - K > > - ? ? - - - K - > @ - A B - ? C B - @ - N ? @

C ? N - - ? @ A N - > - 8 @ N @ ? C ? A B 8 8 > C 8 - ? C N B N K N B

8 A B - 8 A N @ 8 - ? K - 8 B - @ A ? @ - A @ A B - @ - > - @ ? C A - C - @ ? B -

? ? - B 8 > > @ 8 > > - - C C - > - 8 ? A A - @ - ? A B - @ - N ? @ 8 - >

C - - C C N K A ? A - N 8 - A N > - 8 - A @ - A 8 A - K - - @ -

@ B > > - - @ C N N C N 8 N > 8 K A ? C @ ? ? K N K K N - @ - A @ 8 K - C N ? ? @

Page 109: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � ! � � � $ ( ! ! $ ( $ ! $

! $ $ 9 9 $ � $ ! � ( � $ ; � � � 9 9 $ $

� � ( � 9 ! $ � � ! ( � ! � ! ; $ $ ! � $ � $ 9 �

� � ( 9 9 � � � � � � ! � ! � ! $ $ 9 ! $ ( ( $ 9

$ � ! � ( � $ $ ! � $ � �

; � ( 9 9 � ; ! � ( ( � � ( � � ( � $ � � $

� ( ! � � ; ! $ $ 9 $ $ � 9 $ � 9

! � 9 � � � � 9 9 � ! $ � $ $ ( $ � ! 9

$ ! $ � � � ( 9 ( ! � $ $ $ �

$ !

$ ( ! $ $ 9

$ � 9 9 � $ 9 $ � !

9 ( � ! � $ 9 9

$

$ ! ! $

$ !

$ ( 9 $ ( $

$ ( � � ! � � $

$ �

� 9 $ $ � � � � $

� 9 $ ( ! ( 9 9

! $ � � � � ; � ! ! � ( ( 9 �

$ ! ! � � � $ � 9 $ $ ! � $ $

Page 110: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � ( ) ) (

) K L L

P � (

� ) � ( ) � L

� L ( � �

) � � ( L

� �

P ) � � (

� ( � ) �

� � ( ) ) ( ) K

� � � � K ) ) K � �

Page 111: Goldrei.classic Set Theory for Guided Independent Study

� ! " ! % ( ) ( ! % - % ( 0 ) ) 3 - 0 0 ( 0 � - 0 � ( 0 ( ) ( - 0 0 " " � -

" 0 3 � 0 0 ( " � - ( 0 � " 0 ( 0 0 0 "

) " � " - ( " � " " - ( 0 � 0 0 - % % " 0 % - 0 -

! 0 ) 0 " � � " 3 " 0 ) " " - �

( 3

� ( 0 ( 0 3 " � � " " " 0 " "

% ( " ! " " 0 - 3 " 0 0 � " "

0 ! ( ) 3 ! % ( 0 ( 0 ( ( 3 0 ( " " � % 0 � 3 0 � ( " 3

0 ( ( " 3 ( � ( 0 0 ( 0 � 3 % 3 ) " 0 � % % 0 � "

0 � " 0 � % ( ( % " - " � ( - " � ( � - - 0 0 - " % " � 3 ) " � 3 % � (

( % 0 3 ) " ( 3 ( % 0 � 0 3 % � 0 ( "

" ! 0 ) ( � % 0 ) " � 0 ) � ( ) % 3 "

3 " ! ( ( " 3 ( � ( 0 % � 0 ( 0 " � % " � " � 0 "

� 0 % " � � % 0 ) % 0 � " " 0 0 � "

( ( 0 ) " 3 % ( - 0 " � 0 ( 0 0

( % " 0 3 ( ) ( " ( 0 � 0

0 ) % " � ( 0 � " 0 0 � 0 " � 0 - " 0

( - 0 ( � 0 ( " � - ( 0 " 0 0 ) 0 " "

0 � 3 ) " 0 � 0 3 ) - ( " " % ( ( " 0 0 0 " ( " �

( 3 ( " � 3 ( ( - ) 0 % - " - - " ( "

3 ( % - - " � 0 0 % � " 0 0 ( ) 0 3 " 0

" 3 0 " ( ) ( " 3 % 0 ( 0 - " 0 � - % - ( % " �

! " � " � - - ( " - 0 0 � 0

! ( ) ( % 0 ) ! ) " 0 � % " 0 "

0 � 3 ) " 0 0 ( � " 0 0 0

0 - " � " 0 �

0 3 ( " " 0 "

0 -

0 0 ( ( - ! % � 0 " ( 0 � 0 0 "

0 - 0 � 3 " ( % � ) ( � % ) ( � 0 � -

" ( 0 ) ( 0 % 0 ( % " ) 0 " 0 0 - -

( � ( ) 3 0 " (

( ( % � � " 0 0 "

% 0 % - - " 0 " 0 0

3 0 � 0 "

( % � % 0 - 0

0 � 0 ) ( � " " 0 0

Page 112: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � " � � + + / 0 / /

/ � + � 0 / / @ / " � � + @ @ / "

� � � " / @ @ + � " � + � " � / � � @ + � � � 0 � 0 � /

� 0

0 @ @ 0 " @ � / " � � / / / � � + � d � 0 " / �

� � � " � � � " @ " � / � � d � � � � � @ � 0 � + + @ + @ � � 0 � " �

� @ � @ � 0 / " � � d � � d / / 0 � / � " / � � d @ � � � / " � �

� d / d � � � � 0 d � / / � 0 / " @ + + / � / d � � 0 � " @ �

@

� 0 " / / � d �

/ + + / /

� @ 0 0 / " � + d � / 0 � / 0 / / + / 0 / �

� � " / / � 0 / 0 d � + / � + + 0

� @ � 0 / @ + + / � " � 0 " + + � � � 0 � +

@ + � / + � 0 " � + + / 0 � " / " + / �

� � d � " � / 0 � � d / / / � " �

/ d � d d / � d " � � / + � �

� @ /

0 + " d / / /

/ @ � d 0 0 " / + � �

� � � / " d " � / d + /

� / @ � 0 + " / / � / / / d

d + � � � 0 � + � + d / / � /

+ � � d � � � " d + / � / � � 0 � / 0 � / d �

� � @ � / � 0 d @ � / 0 + � " / �

+ " / � d �

/ � � / / �

" � � � d

0 / + � � / + 0 + � / � d

@ � @ 0 � / / 0 � 0

� 0 " � " " @ " � / d � 0 @ 0 0 � @ + / d

" � � � 0 � d + / � " / / / � 0 0 " / /

� � 0 d / d " / 0 � / � 0 � / + /

� " @ � � " + / / / /

� 0 " @ + " / d � / � /

� " + / 0

Page 113: Goldrei.classic Set Theory for Guided Independent Study

� � � & � �

� < � ? < �

� � � � &

� � i

& �

� � & & �

� £ ¥ � � ?

i

i < < & � � i

i &

� � � < � & ? <

� i � �

¥ ? � < � � & �

¥

? & � � ¥ & � � i � & & �

� � � i � ? & & � <

& & < � & �

<

� & � � <

& ? � � <

i � < & i & �

& � i i & & � � &

i < i i

& � < � � � & < �

� i � & � � & � � ? � i ? �

� � & � � & & ? < & i &

& & & � & � � i � & � �

� � i & & � � ? �

i � �

� i ? < ?

& < � � & <

& � < � i & & ? & < �

& � & < < i � �

Page 114: Goldrei.classic Set Theory for Guided Independent Study

' ( ' ) ( ' * ' ( . 1 ' ' * 4 . . ( ; 4 1 * ' 1 . . . 1

' ; H . ( J 1 * ( . ' 1 ( ) ; H 1 ; 1 J J 1 . J ) ; ;

1 ' . * ' . ( ; ) 4 1 * * ( . 1 1 ) ; * 4 . 1 . (

. J ' 1 1 ; . 1

.

) . (

. 1 ' 4 1 . * . ; 4 .

) 1 ; ' 1 . . 1

) * 1 . * J

' 4 ( J * 4 1 ; ; 1 . .

1 . ; ; . 1 . . 1

. . * 1 * * . .

. 4 * .

1

; ) ) 1 ( . .

; ; 4 ; 1 4 1 1 ;

; 1

. 1 * * . . ; . . 1

. 1 1 . 1 1 . * . 1 *

; . . J 4 ; J . .

. . . 1 * . 1 . 1 (

* . . ) ( ) ; ' * 1 1 4 . . 1 1 . .

' * J 4 ) ( . . 1 1 1 1 .

. ( . 1 ; ; ; ; 1

1 . * ( ; 1 .

1 4 1 . . 1 ;

. ' * 1 .

. ( ( J 4 ( . . 1 . 1 1

J J ' 1 * 1 . 1

. ' 1 ) 1 . 4 4 1 *

.

' ; ; * ' * 1 ' ' . . ; ' * . 4 . .

H . ) . * 4 .

) 1 . . 1 ; '

1 J . * 4 * * 1 .

4 1 . J . ; ; . .

1 * . * 4 *

. ; ;

J . * 4 ' . 4 1 * 4 ' 1 . 4 1 . * ' . * ; J ' . ' 1 .

. . 1 ' . H . . J ' ) * ; ' * . . 1 * J 4 ) * ' 1 1 * 1 . ( 1

Page 115: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � ! � � � � � � � � � � + � � � � � �

� � � � ! � ! � � � � � � � ! � � � � � � � � � � + � � � �

� � � � � � � � � � ! � � � � � � � � � ! � � � + � � +

� � � � ! � � � � � � � + + � ! � � � � � � ! � � � ! � � � � �

� � � � � � � � � � � � � � � � � � + �

� � � � + � � � � � � � � � � � � � + � ! � � � + � � � ! � � � � � � ! � �

� � � � � � � ! � � � + � � � �

� � � + � ! � � � � � � ! � � � � � � � ! � � � � � � �

� � � � � � � � � ! + � ! � � � � � � � � � ! �

� ! � � ! � � � � � !

� � � � � � � � �

� � � � � � ! � ! � � � � � � � ! � � � � � � ! � � !

� � � � � � � + ! � ! � � � � � � �

! � � � � � � � � + + � � � ! � � � � + � � � + + � � � ! � � ! � �

� ! � � � � � � + � � � ! � � + � � � � � � + � � � !

� � � � � � � � � � + � � � � ! � � � � + � � � �

� � � � � � � ! � � � �

� � � � � � ! � + � � � � � � � �

� � �

� ! � � � � � � � � � � ! � � � � + + � � � � � ! � � � � ! �

� � � + + � � � � � � + ! � ! � � � � � ! � � � � � � � � � � +

� ! + � � � � � � ! � � � � � � � � � � � + � � � � � � � � �

� � + � � ! � + � � � � � � � � � � + � � � � � � � ! �

! �

� � � ! � ! � � � � � ! � � + � ! � ! � � � � �

+ � ! ! � � � � � + � + � � � � � ! � + � � � � � � � � � �

� � ! � ! � � + � � � � � � � � � � ! � � � ! � � � � � � �

� � � ! � � � � + � ! � � ! � ! � ! � � � ! ! � ! � � +

� � �

� � ! � � � + �

� ! � � � � � !

� � � ! � ! � � � � � � � � � � � � ! ! � � � � � � �

� ! � � � ! � + � � � � � � + � � � � � � � � � �

� � � � � � � � � � � � �

Page 116: Goldrei.classic Set Theory for Guided Independent Study

# % & ' ( ' , & - ' # - / - /

& #

# ( % % - # # # / # , / # % ' , - %

- # ( R % % # '

R # # & ' R # % - %

# % ( ( } - % - % # # ' & ' R # ' , & ' - # % # - ' #

' R ' ' R # % - } # ' # ' # - , # % ' , - %

, % ' R # - # / & # ' / - ' # , - - - , # ,

R - & # & ( % # } # # ' - # ( % % # & & # ' - % } , % # & & # '

R , R ( } - R ( # % & ' # ( % , % - ' - # % # / R - , # % ' , - % /

# # - % ' ( ' - # # R # / # , ' , - % - ' -

R R # & # ' R / # # - / - ' R # , - ( , ' & # & ( % R # ( # # , ( } # % '

# & R ( } } / - % - - % ' ' R # - } } - , % - % # % ' , - %

' R # - - - , # ( / # , ( ' # ( & # # ( % # , ' R # / # , -

/ ( # - / # , R # / - ' R # ' - # , ( } # % ' - / & , % R # - / #

# ' % - } - - ( ' & - # - / # # , } # % ' & ' -

# R ( # ( } / # ( R - % ' R ( ' , } , # & R # & ' ( # # % '

- & # R ( ' , & ( % % , - % # % R # # , (

% - % & ' ( - / ( } }

/ - # ' R ' , , % ' # , ( } # % ' ' - ' & & ' # # '

# % # # - R - ' ' R # & ( # # % , } , # ' / % & - ' ' - # - & '

- % # % # % ' - ' # , ' # - , % ' , - / & - & - & # ' , &

- , % ' ( , } - % - # ' # & # } } % # # - , % - } # & - # - / ' - - %

' , - % ( % ' R # , , & ' - ( #

' R # - R R # } - % &

, & , & # } } # % # % , - % & % - ' - % } - # & # , % # % ' -

( } # ( & - % # # # / - # % ' - % - ' ( } & - ' R # - % , ' , - % ' '

, & ( & - ' , } # & # & ' ( ' # } - % & ' - ( ' - & ' - % # # #

Page 117: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � + � � �

+ 4 � � � � C � � � � � � � �

� � � � � � � � + � � � � � + � �

+ � � � � � � � �

� C 4 � � � + � � � � � � + � � � � � � C � � � + � � � � � � + � �

� � � � + � � � C + � � + � � � � � + � � � � � � � � �

� � + � � � � � � � � � + � � � � � � C �

� � � � � � + � � � � � �

� � � + C � � � � � � � � �

� C � � � � � + �

� � � � � � � � � � � �

� � � � � � �

� � � �

� � � � � � � � � � � � +

� C � � � � C � � � C � +

� � � � + � � � � � �

� � � � � + � �

� �

� � � � � � � + � � � C � +

� � + � � + + � C � � �

� � � � + � � � � � � + � � � � �

� + � + � � � + � � + � � � � � � C � � � C � C + � C �

� � � � � � � � � � � � + � C � � � � � � � � � � �

� � � � � � � C � + � �

� + � � �

� � � � � � � � �

+ � � � � � � + � � � � C

C � 4 � � � � � � � � C � �

� � � � � � + � � � � C � �

� � C � � � � � � + � � � � � �

� � �

+ � + C � � + � C C + � � + � � � � � �

Page 118: Goldrei.classic Set Theory for Guided Independent Study

� " # ' ' . / 1 � � . � ' 7 1 1

/ # 1 / ' 7 A � � ' 7 ' . 7 � 1

7 � 7

� A / ' � 1 ' 7 ' 7 ' . ' A

/ # � 7 \ ' ' . # A 1 " � ' 7 � . 1 ' 7 � ' / . ' # ' � 1

A / . 1 � A " / 7 ' 1 ' / 1 / 1 # / 1 7 A ' �

. # \ ' . / # 7 � � / � 1 / A � �

� / 1 " # 1 � � �

' \ ' �

1 ' # A � ' 7

7 / 1 1 ' � . � �

/ # 1 '

1 � 7 ' 1 A � � � 1 � � ' 7 � A � � A

/ . . ' 7

# � / A

� / A / 1 � A � . �

' � ' 7 1 # ' .

� 7 ' / A / � . . 7 7

� / 1 ' � 7 / 7 ' � / 1 A '

7 1 ' � 7 � A

/ \ 1 " / � ' 1 1 ' 1 1 7 1 ' .

' # ' � '

� ' \ / � � .

' / 1 � � � �

� # 1 ' \ ' . 1 / 7

� \ / " 1 A . / 1 / 7 7 � 7 1 1

1 " / A / ' \ # 7 ' � ' 1 � 7

Page 119: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! � # ! � � $ $ � & ' � & $ & � � # ! � � � � 4 & $ � ! � & '

� � � & � 4 ! # ! � � � !

& � 4 � # � � $ $ � $ ' � & � 4 & # � � & $ � � � $ � �

& � '

! � � & 4 # ! � 4 # � � $ � $ ' � 4 & ' & #

� $ � � � � ' # 4 � '

� # � � � 4 & $ & ' # � � & $ � & 4 � & # ' � ' � # 4 � # � #

� ' & # & � ' 4 ' � � # � � & � � � ' # � � # � � � # � & � ' � ' � ' # & '

� � � � 4 # ' � # � 4 � � & � '

4 & ' � $ & �

! � & � ! � � � � � � ' � # � ! � � ' & ' � 4 � # & � ' � ' # � ' � # � �

� 4 � 4 � 4 � � ' & $ � 4 # & & ' � # ! � � 4 � � & ' $ � $ � $ � � 4 � �

� � # & 4 ! � � $ � � & � � � � # � � # & ! & � # � � � $ � # �

� � # � � 4 # � � $ $ � � � ' ' & $ �

� � 4 � � ' � � $ $ � � � � �

� ' � � & # � � � # ! � & ' & $ � # � � � $ � � � � & 4 $ 4 � ' # � # � � � �

& ' ' & � � 4 � ! � � & ' & $ � � � � ' # � � � 4 & � � ! � � &

& ' � � & � ' � ! � � � & ' � ' & � $ & $ � # ! � ! � ! � � � $ � 4 �

# ! � # ' � $ $ � � � � � 4 � $ $ # ! & ' � & # & ' # � ' � �

# � � # & � ' 4 � # � � � ' � � $ � & $ � � � � $ � � ' � � �

4 � � $ # � � � � � � � � � & & 4 � 4

# � $ � � � � � � & � � & � & ' 4 � � # 4 � � # � 4 & � � # �

� � ' � # & � ' � $ # � $ � ' � � � � 4 & & � � ' & $ �

� 4 � � ' � � � & ' � # & 4 4 & & � ' � & � � � 4 � # $ �

� � � $ � � � � � & � � � # ! � � # � � ' & � ' � # �

4 ! � � � � ! � & ! � ' � � � � ' � 4 � 4 � � ' # � # & � 4 � & ' & � � � � � 4 � ' ' & #

4 & � � � 4 � � � � & � $ � � # � � ' & & ' � # � � # &

� � 4 � � 4 � � � � � � � # ! � � � � � � � # ! � � 4 � $ � & ' & $ � 4 �

� � � ! � 4 & ! & # � � � 4 ! � ! # � ' � $ � � & � '

# & � ' # � � � & ' # � & � $ � ' # � $ $ # & & � � & $ � & ' # ! � � $ $ � ' #

� � � � � � � � � # � 4 � # # $ � # ! �

& � � ! � ! � � # � � � # � � � � # � ! � & & � ' ! � � # 4 ! � # � #

4 # � $ � & � � � � � � � � $ � � # � � # � � & � �

Page 120: Goldrei.classic Set Theory for Guided Independent Study

� $ % & $ ) * ) � 2 ) $ � 2 ) $ 5 $ & ) 2 ) � )

5 * % $ 2 ) W 2 � � 2 � $ � 5 $ ) W * ) * & 2 � $ $ � $ � )

& % � $ 5 $ 5 W 2 & $ * � & ) * ) * $ * � $

* $ $ 5 $ $ 5 ) * * ) ) $ 5 & ) $ $ $ � $ 2 2 � $ � $ ) $ & )

* W $ * & $ � ) * % $ � $ 5 2 � � $ � W $ % % 2 � $ %

* & $ ) W 2 & * ) $ * ) 2

� ) 2 � & 2 5

$ $ � $ 2 ) % 2 $ W * $ %

5 $ ) * ) 2 � ) 2 � & * ) 2 $ 5 $ $ 5 $

2 W $

2 * � � % 2 & $ � ) 2 * % % � 2 � & * )

W $ 5 $ 2 � 5 5 $ � W * $ $ $ 5 5 � $ $ � 2

W $

$ ) 2 � ) � & * * ) % 2 % )

5 � ) W $ 5 $ 2 � ) 2 $ �

$ � ) $ 5 $ 2 & * & W * ) 2 �

* & 2 & � ) 2 � * 5 2 W $ 5 $ 2 & * � )

W $ � & ) * $ 5 $ 2 � ) 2 * %

) 2 � *

� $ 5 & $ % & $ ) ) 2 � 2 � & ) $ & W % % ) * 2 �

& $ $ � ) 2 * % % � ) 2 � & ) $ ) $ $ $ & $ � $ 2 ) % 2

5 2 2 ) 2 & � ) ) W $ $ ) )

� ) � ) 2 � * ) $ * ) ) $ 2 & ) * �

W $ $ 5 * % % W $ $ 2 � 5 2

5 $ W 2 ) 2 $ 2 � ) $ $ 5 ) 2 � 2 $ � 5 2

2 $ � & ) 5 $ � $ % 2

* � % 2 & � ) � ) 2 * % % � ) & )

2 � $ $ $ 2 ) $ � % ) $ 5 $ � 5 * $ & $ �

) $ $ * � $ � $ $ *

2 $ � ) 2 � � ) $ 2 %

W $ $ 2 & & ) * � 5 5 $ % ) ) � $ * % * � * % 2 & 2 * 5 $

5 $ ) W * � * 5 & $ $ $ 5 $ % * � & * * 5 $ � ) % $ ) $ 2 )

) * � &

& ) � * 5 � � * $ � ) * % $ & % ) ) $ ) $ 5 $ $ $ 2 � ) $ 2 � 2

� ) % $ � � % % & $ & & % %

Page 121: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � ' � � , �

� , < = ' > ? � , � � > , > , �

,

� � < � > �

< = � � ? � = � ' � � < ' � � � � >

> , < º > � ? < , � , � � � �

< º � � = = � , � � �

� , , >

< < ' � , � < ,

� , � � � < > � � , < = > � = � , � , <

> < , � >

= � , � �

� , � , � � � < �

º < > , >

� = � � > > � , � > , , � , , < �

� � � < � � , � ' � , � , , � < ?

, < � � � >

= � = >

> ' � , � = � < < � �

> , � , , < = = � , � � � �

Page 122: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � $ $ & & � � � . � � � � 0 $ . : & � $ � &

� . � � . � � � � � . � � $ � � � � � � � � � � .

� . � $ &

$ $ & � � � & � . � � � � &

$ $ . � & � : � : $ $ � $ � $ � � & �

. � $ & � . . � � � � &

. � � � & : � � $ � . . � � � 0 � � 0 . $ � : . � � 0 �

& � & � � � $ � � � � � . � � � � � � $ & � � & � � � � �

� & � � � � : 0 � � � & $ � � . � $ & � � & � � .

� & $ & � � � � � � � : . & � � � � � �

� � . � � : � & $ � : � . . � & . � � : � � � � $ � �

� � � . $ � & : $ . � � � � � 0 � � � $ � & : � . . � � � $ &

� � : & � & � � . � � . � : � � � � � . . � � . � � � � &

0 � � $ & $ � � . . $ & � � � � : � �

� & � � � � . . $ & � � � . � $ � . � � $ . � . : � � $ �

� : � � � � & & � $ & & : � & � � $ � & $ � � � � 0 � �

� � & � � � � � � $ � � � $ &

� 0 � $ . � & $ $ � � $ � & � & � � � & �

� � � � � $ & � $ & � & . � � � � &

: � � : � $ � & � &

: & � & � � � �

$ . � � � . � $ & � � & � � : $ � & � & � � � $ & . �

� � � � � � � $ � & � � & $ � & � � � $ $ � � : & �

: & � � $ � � & $ $ & $ & � $ � � & � $

� � � � � � � � � & $ & & � � &

� 0 � � . . � � $ � $ &

� . �

� � � � � � � : � � $ � � & � � : . & � �

� � � � . � � � : � � � � . � . � � � & : $ & . � : � & � � $ � � �

� . � � � � � $ . � $ � : . 0 � � 0 � 0 � & : � � �

� � � � � . � � � � � $ & � . � � . � � : � � & � & � �

: & � & � � & � � � & � 0 $ � & � & .

� 0 $ � � & � � � . � $ & : � $ � $ & $ � � � : � � . � � & � � �

Page 123: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � �

7 � � 8 � � � �

8 � � � � � � � � 8 � � � � � 8 � 8 �

� � � 8 � � 7 � � � �

8

8

� 8 8 � � � � � �

� � 7 � � 8 � � 7 8 � � � � � � � � � � � � � � � � 7 � � �

8 � � � � � � � � 8 � 8 � � � � � � � � � � � � �

� � �

� � � � � � � 7 8 � 8 � � � � � 7 � � 8 � � � � 7 � � � � � � � � � � �

� 8 � � � � � 7 � � � � � 8 � � � � � � � � � �

7 � � � � � 8 � � � � � 8 � � � � � � 7 � 8 � � � � 8 � 8

� 8 � 8 � 8 � � � � � 7 � � � � �

� � � � � � � � � � � � � � � � � � � � � � � 8 � � � �

� � 7 � � � � � 7 8 8 � 8 � � 8 � � 7 � � 8 � � � � � � 8 � � � 7 �

� � � � 8 � � � � 8 � 8 � � � � 8 � � � � 8 � � � � � 7 � 8

� � � � � � 7 � 8 � 8 � � � 8 � � � � 8 � � � � � 8 � �

� � 8 � 7 � � � 8 � � � � 7 � � � 8 � � � � 7 � � � � � �

� � � � � 7 � 8 � � � � 7 � � � � 7 � � � 7 � � � � � 8 �

� � 7 � 8 � � � � � � 8 � � � � � � � � � 7 � � � � � � � � � � � �

� 7 � � � � 8 � � �

� � � � � � 7 � � � � � 7 8 � � � � 7 � � � � � � 8 7 � � � 7 � �

� � � � 7 � � � � 8 � � � � � 7 8 � � � 8 � � � 8 � �

� � � � � � 7 � � 7 8 � 8 � � � � � � � � � � 8 � � � 8 � � 8 � � � � �

� � � � � � � � � � � � � � � � 7 � � 8 � 7 7 8 8 � � � � � � � � �

� � � � � � � � � 8 � � � � 7 8 � 8 � � � � � � � � � � � � � 7 � � 8 �

� � � � � � 8 � � � � � � � � � � 8 7 � � � � � � � � 7 � � � 8 � � �

� � 7 � 7 � 7 � 7 � 8 � � � � � � � � � � � � 8 � 7 � � �

� � � � � 7 � 7 � � � � � � 8 � 8 � � 8 � � � � � � � 8 � � 8 � 7 �

� � � � � � 8 � 7 � � � � � � � � � � � � 8 � 8 � � � � �

� 8 � � � � � � � � 8 � 8 � � 7 � � � � � 8 � � � � � �

� 7 � 7 8 � 7 8 � � 8 � � � � 8 � � � 7 � 7 � 8 � �

� � � � � � � � � � � � 7 � 8 � � � � 8 � � � � 8 8 �

� � � � � � � � 8 � � � � � � � � � � � � � � 8 � � 8 � � � �

� � � � � � � � � � 7 � � � � � � � � � � � 8 � � � � � � 7 � � � 8 � � �

� � � � 8 � � � � � � � � � � � � � � � � � � � � � � 8 8 � � 8 �

Page 124: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � ! � � � � $ � � � � � � � � � � � � � � � � �

� � � ! � � � � � � � � � � ! � � � � � � � � � � � � � � � � � � � �

! � � $ � � $ � � � � � � � � ! � � � � � ! � � � � � � � � � � � � � � � � � � ! � $ �

� � � � � ! � � � � � � ! � � � � � � � � � � � � $ � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ $ � � � � � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � ! � � !

� � � � $ � � � � $ � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � $ � � � � � � � � � $ $ � �

� � � � � � � � ! � $ � � � � � � � � � � � � � � � � � �

� � � � � � � � ! � � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � ! � � � �

� � � � � � � � � � � � ! � $ $

� � � � � � $ � � � � ! � � � $ � � � � � � � � � � � � � � � � � � �

! � � � � � � $ $ � � � �

� � � � � � ! � � � ! � � � � � � � � � � � � ! � � !

� � � � � � � � � � � � � � � � � � � � � � � � � � $ $ � � � � �

� � � � � ! � � � � � � � � � � � � � ! � � � $ � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � $ � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � $ � � � � �

� $ � � � � � � � � � � � � � � � � $ � � � � � � � � � � � � � � � � � �

� � ! � � � � � ! � � $

� � � � � � � � $ � � � � � � � � � � � � ! � � � � � � � � � � � � � ! � � � �

� � � � � � � � ! � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � ! � � ! � � � $ � � � ! � � � � � � � � � � � ! � � � � $ $ �

� � $ � � � ! � � � � � ! ! � � � � � � � � � � � � � � $ � � � � � � � ! � $ � � ! ! � � � $ � �

� � � � � $ � � � � � � � � � � � � ! � � � � � � � � � � � $ �

� � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � $ � � � � � � � � � � � �

� � � � � � � � � � $ $ $ � � � � � � � � � � $ $ � � ! � $ � �

� � � $ � � � ! � � ! � � � ! � � � ! � � $ $ � ! � ! � � � � � � � � $ $ $ � �

� � ! � ! � � � � � � � � � � � � $ � � � � � ! � � � $ � � � � ! � � � �

� � ! � � � $ � � �

Page 125: Goldrei.classic Set Theory for Guided Independent Study

! % ! ' 8 =

8 = 8

= i m m = q =

8 q i m m = '

i ' = ' '

8 i m m i ' = = ' ! '

% % 8 = i = i m 8 8

q = 8 q q = 8

i m

8 ! m ' m m ! = 8

% = ' 8

8 8 q = ' m ' =

8 8 q % ! 8 m i = ' 8 m ' =

8 = ' 8 ! ' 8 8 i ' 8 = m

m = ' 8 ! 8 m m ' = = 8 = 8

8 8 = ' q m

' m 8 ! = = =

= i 8 8 q 8

8 q = = ' % 8 8 ' 8

i ' ' q = ! 8 !

= ' 8 ' m =

= m =

m m 8

! m

! 8 8 q m = m 8 q 8 8

= i ' = q ! = m 8 8

8 ' q 8 ' 8 = q = ' i = m 8 q m

8 q m ! 8 i q ! 8 m 8

8 m 8 m 8 ! 8 m = = 8

i 8 q = = = 8 ' 8 '

Page 126: Goldrei.classic Set Theory for Guided Independent Study

� � � � % % %

; < � = > = ? � � = % B % B % ?

� ? � � = ; B ? = % B = %

=

; � ; > ; � B = > ? ? ; � ; B � % ?

; % � ; = � � � > r B ? ; B � > ? ; %

� � = � � % � B = r � > ; r ; > = � % � ? = % � ; %

� > ? � � � ? % B ? B ? ? � r > �

� � � � � ; > ? ? � ; � % % r ; ? B ?

r � r �

> < % ; > ; B > B ? ? = ; %

r � < � � > = � > B � = ; % %

� < � > B = % B B

B = � % ; ; % % ; % %

B

B

� = � � ; % % ; %

% � r � % ? = r B

; ; < B B � = % ; ? = B ?

B B ? r ? ?

� � ; ? ?

B ; � � ? B = ? %

B % r > ; % ; B ? r r

< � = > ; r ? = B % % B � %

� � ; � % B ? ? % ; % B = ;

< � B ; B ? ? r % % B =

B ? B � � ;

= B ; B % = B B ? ? ? % r % B B

= % r

% r < � � ; B B r B

B < > = ? % B � B ; � %

� ? � � = % B B = B B � > ? � r % = %

B B = ; B > % > > = � r % % r ;

% r � � � % = > B > ; ; = % ? = ; %

� � � % ; % B =

Page 127: Goldrei.classic Set Theory for Guided Independent Study

� � " � # � $ % ) + , � # � $ # % % � " 4 � 4 + , 4 ) � ) � � + $ #

% " 4 4 > � $ �

" 4 # + % , 4 � � � + $ # % 4 + � � � + % " � > � , 4 4 � > + , �

# � % " � + � 4 ) % 4 + � � � + $ % % + $ 4 , , + > % � �

$ " # , # + % $ , )

� � $ " � � % 4 ) % + % + � + $ , % # , $ 4 , + 4 # � $ % 4 +

% % , + 4 4 ) ) % > � > � + � , # � 4 % � % # �

$ " 4 $ 4 4 , ) � ) � � $ % 4 $ # � , 4 ) > % � � , 4 + % � $ "

+ $ % + % # � # #

� 4 $ 4 # # � % , 4 ) � ) � � + % , $ $

� , % , ) � $ % 4 � , $ % % # ) # % 4 +

� ) � � % , , # ) # % + % + � + $ , % + 4 > � % + % $ 4 4

+ � > % > 4 + " , + % 4 $ " , % ) # $

% � + 4 + $ + , $ % ) + � # , + � 4 ) � , ) � 4 � # " %

4 � # $ " � % > � � 4 ) ) � # � % � 4 ) #

$ � � # , ) ) % + % % 4 � 4 # �

4 ) % % $ 4 4 , + 4 + � > 4 , > $ % % # � + � 4 $ � # 4 + , % +

, , # + % $ % + % + � " � + � + % ,

+ " � # � $ 4 % � 4 � 4 % + + � + , % + > % +

$ � # � $ " 4 $ 4 + � $ " � , + � > $ , % + , # % + % + � + , % +

> % $ %

% # 4 $ � % > + # ) � � � � % + � > � > � + , , + % ) ) %

� $ � 4 # $ � 4 ) > $ , 4 ) ) % > � > � + ) � � 4 + %

� $ # # � $ � , # 4 , $ % � 4 , , $ $ � % ) ) % , +

$ , % + ) � >

% > 4 ) ) % >

+ � % " , 4 > $ , 4 % > � 4 # # % 4 $ , # � % �

% > ) ) % > %

) � " + > �

, + 4 > , 4 ) % > � �

4 +

% , + $ � > # , + > $ , 4 ) ) % > � �

Page 128: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � �

� � � � � � � � 3 � � � � � � � � 3 � 8 � � � � � � � � �

� � � 8 � � 3 8 � � � � � � 3 � 8 � � � � � � 3 � � � � � �

� � 3 � � � � � � � �

� � � � � � � � � � � � � � � � 3 � � 8 3 � � � � � � � � � � � � � � � � � � � � �

8 � � � � � � � � � � � 3 � � � 3 � � � � � � � � � � � � 8 � � � � � 3 � �

� � 3 � � � 3 � � � 8 � � � � � � � � � 3 � � � � � 3 � � � � � � � �

3 � � � � � 3 � � � � � � � � 3 � � � � � 8 � � � � � 3 �

� � � � � 3 � � � 3 � � � � 3 � � �

� �

� � � � � 3 3 � 3 � � � � 3 � 3 � 3 � 3 � � 3 � � 3 � 3 � �

� � � � 3 � � �

� 3 8 � � � 3 � 8 � � � � � 3 � � � 8 � � � � � � � � � � � � �

3 � � � � � � � 8 � � � � � � � � � � � � � � � 3 � � � � � � � �

� � � � 3 � 3 � � � � � � � � � � � � � � 3 � � � � � � � � �

3 8 � � � � � � � � 3 3 � 3 � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � 3 � � � � � � 3 � �

� � 3 � � � � �

3 � � � 3 � � � � � � � 3 � � � � 3 � � 3 � � � � � � � � �

� � � � � � 8 � � � � 8 � � � 3 � � � � � 3 � � � � � � � � � 3 � � � �

� � � � � � �

� � � � � 8 � � � � � � � � � � � � � � � � � � � � � � � � � � 3 � �

� � � � � � � � 3 � � � � � � � 8 � � � 3 � � � � � � � � �

3 � � � � � � � 3 � � � � � � � � � � � 3 � � � � � � � � 3 �

� � � � � � � � � � � � � � � 8 � � � 8 � � � � 3 � � � � � � � �

� � � � � 3 � �

� 3 � � � � 3 8 � � � � � � 3 � � � � � � � � 3 3 � � � �

� � � � 3 � 3 � � � � 8 � 3 � � � 3 � � 3 � � � 3 � � � � � � � 3 � 3 � � � 3 �

� � � 3 3 � � � 3 � � � 8 � � � � � � � � 3 � � � 3 � � �

� 3 � � � 3 � � � � � � � � � � � � � � � � � � 3 � � � � � �

� � � 3 � � � � � � � � 3 � � � � � � � � � � 3 �

� 8 � � � � � 3 � � � � 3 � � 3 3 � � � � � 3 � � � 3 � � � � � � � � �

� � 8 � � 8 � � � 3 � 3 � � � 3 � � 3 � � � � � 8 � � � � 3 � � � � � � � � � � � 3 � � 3 �

� � � � � � � � � � � � 8 � 3 � � � � � � � � 8 � � � �

� � 8 � 8 � � 8 � � � 8 3 � � � � � 3 � 3 � 3 � � � � � 3 � � � � � �

� � � �

Page 129: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � $ � & � & & - � $ 0 � 0

� � $ � � 0 � $ � � � � & � - � 0 $

� � � - 0 $ & � 0 � - $ � 0 � - - 0 $ � � $ � 0 � � &

& 0 � � � � $ & � 0 � $ � � � - � � $

� - $ - & � 0 $ � � - � - - $ � � $ � � � �

$ � � � & � & 0 $ � � � � � - � � 0 $ � � 0 � - $ �

� � - � � � 0 � � - & � 0 � � � - 0 � �

� � - $ - - � � - � � � �

� 0 � & � 0 �

& � 0 $ � 0 $ 0

$ � & 0 $ � & � � �

� � � $ � � & � 0 $ � &

$ 0 $ � - $ $ $ � � & & $ � � - � - � � � - $ �

0 - � � � � � $ 0 � � � 0 � � 0 $ � &

� & 0 � � 0 $ � - � � $ - $ $ 0 � $ � & 0 $

� & � �

� & � � � & 0 � � 0 & � � 0 � 0 & 0 0 0

$ 0 � � � � � 0 0 0 $ �

� � � � � - - 0

- $

0 & � - � & � & � & 0 - 0 $ $

� � $ 0 $ � � � 0 � $

� � - � � & � & � - 0 0 0 & �

� � $ - & & � � � & � 0 $ 0 $ - � - � � & � �

$ $ 0 � 0 � $ � 0 $ - & 0 � 0 � 0 &

0 $ - $ � � � � 0 � 0

- � - � 0 � - 0 $ 0 � 0

� �

- $

& �

& - � � $ � 0 $ - $ � - - � 0 � � � - � � � 0

- � � & � - � � $ 0 � 0 � - 0 & � � � $ $ � � � � - 0

$ � � - � - $ 0 � � 0 � 0 � $ � � - � � � & 0 $ - 0

& � � � � � � � � 0 $ � - � 0 $ - � & 0 $ 0 � � 0 $ 0 � &

� � � & � � - � � 0 � 0 $ � 0 $ $ �

� � � � � - 0 0 � � 0 $ 0 $

� � & � 0 $ - $ 0 � - $ $ $ 0 �

0 � 0 $ � - & � � $ � $ � $ � & 0 � $ & � � 0 &

- � - 0 � - � � � & & � � � 0 �

� � � $ 0 � � 0 $ � - & 0 $ - � 0 $ � � & � 0 $ � - 0 � 0

0 � � � 0 $ � - � 0 $ � � $ � $ - $ 0 � � - $ 0 �

Page 130: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � " � % � & " * & � � & - � " % *

� � & - � * � - � " & � � � � � " � " * � * " - � " * " - - * & � -

* & � � & � & � � � * " * " - - * & � � - * & � � & � & � � &

� % � � � * " � � � � % * % * & � � * & � % � " & � � % � & * " - " � � �

* & � " � - * & � � & � � � � � % " & � & � � * � * � � * " - � � � �

� & � " � � � " - - * & � " � * & � � & � & � � � � " � � * � � � �

" � � � � � � " " � * � � % � � & * � * � & � � � & - � " � � �

% " * " - � - � � & �

� & * � � � & - � � � � � � � � � - * � � * - � � * � � & � " *

" � � " � � & � & � & � � � � � � � � " � * � & � � " * � � & � * & � �

- * & � " - * � � & � � � � � % � � % " � & " * & %

� � � & * & * � % � % � � * � � " - � � & " � - " & * � �

" & � � � & * * � " - * & � " � - * & � � � & � � � � � � � �

% � % " � � & � � � � � * * - * & � " � - * & � � & � & � " � & * � - " &

� � �

% � & � % � � � " � � � - � � & �

% % " �

� � � % " % " * & � % � � � � �

* & " & * " & " � " - & � � " & � % � � � � � � & * � � & � �

� % � � � & � & - - % � � � " � � % � & � % * � & � " * & " - -

� " " � � * - & � " � * & � � & � & � � � � �

* - � - * " � - * & � � & � & � " � � � �

% * & � � � & � � * & � � � � � & - � " � & � " * &

" - � - � � & � " & * � " " *

� % " - - - � " � � � � � " � � " � " - � & � � & � � � � � �

� � � * � " � � % � � & � * � * � &

� & � � � & � � � � � % � � � � * " - - � � � % � � � � � * - " � * & �

� � & - � " * � � � " - - � % " - - � - - � � % � � � - � � � � � " &

" � � * � " & � � " & � � " � & � � � � � � & � � &

� � & � � � % � & " * & � � & - � " � % � � � * " � & � & �

� * � & � � " � & � & � & � �

Page 131: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � �

? � � � � � � � � ? � � ? � �

� ? h h � i � � m � � � h � � � � � i ? � � ? m � �

? � h � � � m � � � ? � m � ? � � � � � � � � ? � � i � � � i

? � � � � � � � ? � � i � � � i � ? i � � � m � h h m � � m � �

� � � � � � � m � � � ? � ? � � � � � i � � � � � � � m � ? � h � h � � � � � ? � �

? � � � � � � m � � i ? ? � h � � � � ? � � ? � ? � � � � � � m � m � �

� � m � m � � � � � � ? � � � � h h ? � � ? � � � m � � m � ? �

? � � � ? ? � � m � � ? � h h � � � � i � �

? i � � � � � i � h � � � � � ? � � � � � � � � � � � ? �

� � � � � � � � m � m � � ? � � � ? � � ? � ? � � � � � � m

? � � m � � m � � � � i � i � � h � � � � �

� m m � �

� �

� � � m � ? � � ? m � i � i ? � m � h � � � � � � i � h ? � � � �

� � � m � � � � � � � � � i � ? � � � � � � � ? �

� � ? � � � m � � � � i � ? i �

� � m � h h h � � � m � ? � � m � � ? � � � � � � � � i ? � � � i �

m � � � ? � � i � � h � � ? h � h � � � � � � � � � i ? i �

i � � h � � h h � �

� � � � m � i � � � m � � i � � m � � m � � ? � � � � � � �

� � m � � � � � i � � m ? � � � � m � � m � � �

� m � i � i � � h � � � � � ? � � �

� ? � � � ? � � ? � � i � ? � � � h � � � � � � i � h ? � � � �

� � � � � m � � � � � m � � � � m � � � � ? � ? � � � � �

� � � � � � � i � ? i �

� � � � � � � � m ? i � � h ? h h � � � � � ? � m � � � � � � � ? � � h ? � ? � h � � � i

? � � � � � � � m � � � i ? � � h � � � i ? � � h ? � � m � � i � � � m � � � � �

h � ? h ? � � � m � ? m � � � i ? � h � ? � ? � � h � � i � � m � h h m � m � �

� i � � m � ? m � � i ? � ? h � � � � � � � m � ? � � ? � � �

� ? � � � � � � m � m � i � m � � � h i � � h � � � m ? � ? � � � m � � i �

� h ? � � � � � � � � � � m � i � i � � � ? � � h � � � � � � � m � � � � �

h � ? h ? � � m � � � m � ? �

Page 132: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! � � � # & � ! ( & ( � * � � ( � � � � ( � * � # * ( � � � �

� ( � # � � � ( � # � ( * �

� * � � ! � ( * ! � & ! � � # � � � � # # � # � � * � � * ! # � #

� � � ( * # � ( � # ( � � � * � � ! # ! * * � ! �

� ( * # � ( � # ! # ! # ! � � ! & & � �

# � ( * ! � � � � � ( � � � * � � � � ( � � � ( � ! # ! # � ( !

& � # � � � � � ( # ( � � � # � � ( # ( & � � � # * & � ! ! � ! � ( � * ( � �

� ! # � ! � � ! � ( # * � � # � * � � � ! & ( # ( � ( � � � ! # � � � * � ( # # ( � �

* � � # & ! � � � ! � � � ! & & * � � � * ( # � ( ! � � & ( # � & � � ! #

� � * � � � � � ( # # ( � # � * � � ! � ( # ! & * # � � # ! & * ( ( * � �

� � ( # ( * � � � ( ! & * � ! � # � ( # � � * � � � � ! � � � ! & � � ( * ! # � � �

* ( ( & � � �

� � � ! * � � � ! & � � ( ! & & � � � � � � � ! * ( # � � # & � !

� ! � ! # � � � � # � � � ! & # � ! � � � � � # ( � � ! � (

& ( # � ! � � � � � # ! # � � ! & & ( � � # ( # � * �

( * ! � ! ! � ( & � � � � # ! # ! # ! � ( ! & &

� � � * � � � # * & � � � * � � � ! ( ! & � & � � #

# * � � ! ( ( * ! � ! � ( ! & & ! ( ! � � � � ( � !

� � & ( # � ! � � � � � ( * ( # � & � ! � � � � � � � � # ( # � � ( � * � ! & � *

� � � ! � � ! ( � ( # ( � ! & ( & ( ! ( � ( � # � ( � ( #

� � # * & � � ! � * ! ( ! & � � � � ( � ! &

( � * � � & � ! & � � � ( # � � ! * ( � ! & � ! � ( ! & & � � � � � � � �

( � � ! & � � # * & � ! � # � ( # � ! � ! # ( # ! * ! # � �

� # # � ( � # � � � � � * & � � � � ! � ( � ( # * ( � ( � # � ( �

� ! � � � ! & & ( � & * � ( * ! � � � � ( � ! & ( ! # � & (

� � ! � � # * & � ! * ! � # � � ! & � ( # ( & � � � ! & & ! � � ( ! & & � � � � � �

( * � ( ! & � # � � � � � � � � & & � # � # � � ! & � ( # ( & � *

! � * � � � ( ! & � � � ! ! ( # ( ( * ( & ( �

� � * � � � � ( ! * ! # ! ( # * � ! � � � � � ( * � ! ( #

� # � # ( # ! * ! � � � � * * � �

� � � � � � � ! � � ( ! & & � � � � � � � � ( * ! ( ! & ! ( #

� � & & � # � � � * � * # � ( ! � � * � � � � � ! � � ! � ! & � # * � � # � �

� � � � ! * * ( # � � # * & � ! � � � � � � � * ! � ! * ( *

� � � � � # � � � � � * � � � # & � ! � �

Page 133: Goldrei.classic Set Theory for Guided Independent Study

� � � $ % & ' ) � ' % � & $ * � ' � � * ) ' ' ) � %

% ' � $ * = ' ) > = ? � ) & $ & ' % & $ = % ? * � ? ) ? � � $ � ? � $ % )

> = ? � ) & $ ? ' % $ = % ? % ? = � $ * � � = ' ) > = ? � & $ & ) ? ' % & ) $ = % ?

' * $ * $ % & ) > >

? � & � ) & � ) ' � % & ) � % � & $ � � > � ? = $ * �

% $ % � $ % & = � & � ) > ? � & ' ' $ � = & ' ) > ' = ? ) ? � ? = ? $ * �

& � ' > $ % ' = ? $ � � $ & $ � ' ) > ) & $ % � � ' ) > % $

= ?

) * % $ * $ = ' ) > = ? � ) & & ) ? ' % & ) $ = % ? % � & = ) ? % ? > =

> >% &

� ? � & % � $ * ) $ $ * � � ' ) ? $ = % ? ' * $ ) $

% & ) > >

* ' % $ * � &

% & > >

* � & � �

Page 134: Goldrei.classic Set Theory for Guided Independent Study

) * / * 0 1 2 3 2 / 7 2 / 1 9 3 7 2 7 9 2 3 2 1 * 7 7 * 2 / 0 7 2 / 2 9 *

0 * 2 2 7 2 3 2 7 9 2 2 9 3 3 3 7 2 ) 0 * 3 9 * 9 0 7 9 3

7 ) * / 2 W * 2 3 * 3 1 1 2 7 * / * / * * 3 7 9 2 3 2 0 3 9 2 2 / 9

1 * 9 * 1 2 7 0 9 3 9 3 / W 7 2 ) 2 /

9 3 2 1 1 9 3 2 1 2 W W

3 2 W W

2 / 0 / 9 2 1 1 2 / 7 1 2 1 1 * 7 2 / * 0 W W 7 * / 2 9 7 ) 0 7 7 *

/ * 1 W ) 2 W 2 / 1 * 7 1 ) 9 * W * 3 9 * W 3 2 9 3 1 / 2

3 0 * 7 2 * 7 * 3 9 7 7 2 ) 9 * W * / 7 0 9 3 1 * 7 1 2 2 9 3 2

* * W 0 9 3 7 * 1 7 / * 2 1 ) 1 * 7 7 2 1 * 2 7 ) * * W * 2 9 3 7 1 3 7 2

* * W 2 * 7 * 2 3 3 9 * 0 1 * * 3 W * 1 2 3 7 / 2 * 9 0 W

* 0 1 0 7 3 7 * 3 9 7 9 * 3 7 / W 3 / 1 3 7 2 7 9 7 2 7 3

7 2 1 * 7 ) * 3 0 7 / 0 W 3 * 1 * 9 3 * 0 3 1 *

2 3 9 3 2 9 * * 9 2 0 3 9 * * 3 2 3 9 9 ) * 9 /

3 0 / 0 W 2 / 9 3 0 3 * * 2 * 1 / 9 * 2 * 1 * / 2 9

3 * 2 0 3 2 W 0 * W 9 3 ) * 1 7 * 1 * 1 9 * 3 / 0 W 3 * / 1

) * * 7 2 0 1 * 7 0 * 9 1 * 7 2 7 2 * / * 3 7 1 2 / 7 1 2 3 * /

7 2 * 1 / 9 * 1 9 * 1 2 * 0 3 7 2 * 1 / 9 * 0 1 2 2 / * / 9 3

1 * 1 / 7 ) * / 2 / * 0 3 2 / * 3 2 * 7 ) 2 / 1 2 3 * / 9 ) 0 3 0 / 7 ) * 9

2 * / 0 7 2 3 1 2 0 9 2 W 7 9 W 9 0 9 2 3 0 3 * 2 3 * 3 9 0 7 9 2 3 0 / * * 9 3 9 ) 7 ) *

0 W / 7 ) * 7 9 2 3 / 0 W * / 1

3 7 7 * / * 1 ) W W 2 2 7 2 * 2 7 * 2 / 2 0 / 3 W 3 * /

1 * * 3 ) 0 * * ) 0 W W W 2 2 7 ) 2 / 2 2 / 9 3 0 W 3 / 0

W 9 3 * 7 * 3 7 ) 7 2 2 2 * 7 2 / 2

0 / * 1 * 9 0 W / 7 2 2 / 9 3 0 W

1 7 * 9 3 7 * / * 7 9 3 0 / 0 W 0 / * 1 2 * 9 3 2 2 9 7 9 3 3 9 7 * * 7 1 W * / 0 W W

7 ) * * 3 9 7 9 2 1 2 0 3

1 * 7 9 1 1 9 2 * 9 * / * 1 * 3 2 / 1 2 *

0 0 3 * 9 0 * 9 7 * / * 9 3 ) * 2

2

W * 0 / W * 0 ) 9 3 9 7 * / * W 9 1 9 3 3 9 7 * 7 9 1 0 1 2 3 9 3 / 9 1

2 * 7 9 2 3 * W 2 9 3 7 * * 2 3 ) 2 W * / 9 3 W * 2 * * 0 7 W * 0 7

2 3 * 9 3 9 7 * 1 * 7 3 0 * W 0 3 1 2 / * 0 3 0 7 / 0 W 3 * 7 9

3 7 * * 7 9 1 9 3 7 9 7 * W 9 * / 7 ) 0 3 * 3 9 7 * 1 * 7

Page 135: Goldrei.classic Set Theory for Guided Independent Study

� � ! " % � % % ! " , � � , % � " , � � . � % " 2 2 ! , % � ! �

� . � ! % ! . A � � � , � � , % � % . ! 2 . , % , � , 2 % " %

A � % ! ! % A % 2 ! � � % ! , � % " " % , � A � � � ! A ! A

� % ! " % " % , 2 % , � A % ! " ! % % � � 2 2 % . . " % 2 . 2 , � A % % � , , ! �

! % � � " � . � % " �

, 2 2 % � � � 2 ! � % % . % % �

2 % � % 2 % % � % , ! � " % � , � A , 2 � % . % � ! " ! . % 2 �

A % , � A , A % ! % % � % � ! , , A % 2 % " ! . % 2 ! " , �

, A , � % � � 2 A � , , 2 � � 2 % , A , � � . � 2 2 ! " 2 % %

� � % " ! � , 2 " ! . % � 2 � " � 2 % � % " � � . . � � " � . % " 2

� % ! � % � % % , � % 2 % � % 2 , , � � 2 , � % " � , %

% � " % ! ! " % � , ! � 2 ! ! 2 % ! A % " , % % % . " % � 2 ! � � . %

� . % " � , % % . % A , � , 2 A � , � , , % . % . % % � 2 2 !

A � % , 2 , � % . � 2 ! % , � % � % " � . % , A % % ! � % !

A ! ! 2 % � 2 % ! ! 2 A � . 2 2 2 ! % , ! ! A ! , % , . " % , � %

, 2 2 , ! �

� � ! " 2 % ! " " � ! , , � 2 , A ! % � % % 2 % ! %

, ! ! A ! , % A " . 2 " " , 2 , � , % . , , , � � , % 2 % 2 � " % , � ! . %

! , , 2 2 � ! , , � , 2 A % % % � A ! 2 % " , A 2 % � � A ! %

A , A ! � A % % ! " , A , 2 " , A % " % % � , A ! ! � % � � % " , %

� � ! � � ! " 2 ! 2 % � . % � � 2 ! % ! � " ! % " 2 , � . " % 2 . 2 ! "

� � � % % " % " % A % % � % " ! � , ! � � . � % 2 � � � " � .

� � A % " % � " % � � ! " % , " � ! � � . " % � . � % 2 � � � , � � . "

� % , % . � , , � " ! % " % ! A % % 2 % " % ! � � .

A � � � " . % " � , , 2 � ! � A % % � � � ! , 2 , � , " "

, ! � � . % " % , 2 � , ! � � , . % % % % � � � ! , 2 , � " , ! � � . 2 % " % 2 � �

, " " � , ! � � . 2 A ! . A % " % % � � ! % , " " � , ! � � . 2 � " � , ! � . %

" 2 A , � 2 ! , % ! � � % % � � ! � % 2 % � , � % 2 � % �

! % . % % � 2 � 2 ! " % % . % % � � � � ! % " � , � % , �

% � % 2 % , ! �

2 " % � . . % " 2 % A � A % , ! 2 � � 2 % " , 2 % % 2 %

� � % % " � . % � � � , , 2 � � " . ! � A � , 2

, % " � � A ! . % 2 , � % , % " ! 2 2 2 % !

A % ! A % "

% , % " � 2 ! , �

! , 2 ! " ! % � % ! " % % � , � ! �

� � ! " � . . , % % ! . 2 " % . 2 � A � 2 , % " A � � % 2 % � !

% � 2 2 % ! ! 2 � A % � , , � � % A ! � % . % � " % ! %

2 % A % ! 2 , % " A � � � 2 ! % ! � % 2 , , 2 . % � " A � % , � % � , % "

A � � ! % � � ! � , 2 � " ! % " 2 2 % , � A % ! % ! 2 % 2 !

2 % 2 A , 2 � ! A % % � � ! � % . % , A % % � � � A % � 2 % !

A % 2 % � " . � " 2 % 2 � 2 ! 2 ! . ! � . 2 � A � A � 2

% . % % � 2 2 % . % % � � � , 2 % � % � �

Page 136: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � % � � � � % � � � � � � � � � � � � . � � � �

� � � . � � . � . � � � � � � � � � � � � � � .

� . � � � � � � � � � � � � � � � . � � � � � � � � � � . � � � � � . � � � %

� � . � � � � � � � � � � � � . � . � � � �

� . % � � � � � � � � � � � � � � � � � � � . � � . � � . �

� � � � � � � � � % � � . � � � � . . � � � � . �

. � � � � � � � � . � � � � . � � � � � . � � � �

� . � � . � � � � % � � . � � � � � � � . � � � . � � .

� � . � . � � � � � � . � � � �

� � � � � � � � � . � � � � � � � % % � � % � . . � � � � � � � � � � �

� � � � . � � � . � � � . � � � � � � �

� � � � % � � � � � � � � � � � � . � � � % � � . � � � � �

� � � � � � � � � � � � . � � �

� � � � . . . � � . � � � � . � � � � � % . � � �

� . � . � � � � � � � � � � � � � � � . � � �

� � . � � � . � � . � � � % � % � � � � � � � � � �

� . � � � . . � � � � � � � � � % � . � � � � � � % � � . % % � �

� � . � . % % � � . � � � % � � � % % � . �

� � . � � � � � � � � � � � � � % � � � � � � . � % � � � � �

� � � � . � � � � � � � � % . � � � . � . � � � � . � � � � �

� � . � � � � � � � . � % � � � � � � � � � � � �

� � � � � � � � � � � . � % % � � � � � � � � � �

� � � � . � � � . � � � � � � � . � � � � � � � % % � . � � . � � � % �

� � � � � � � � . � � � . . . � � �

� � � � � � � � � � � �

� . � � . � � � � � � � � . � % � � . � . � �

� � � . . � � � % � . � � � . � � � � � . � � . � � � � � � � % � .

� � � . � � � � � % � � � � � � � � � . � � . � � . � � � � � � � �

� % � � � � � � � . � � � � � � � � � � � . � � � � % � � �

� � � � � � . � � � � � � � . � � . � . � � � � � . � � � � � � � . � � �

� % � � � � . � . � � � � � . � � � � � % � � � � � � . � � . � � � � � � � � � � �

. � � . � � � � � � � � � � � � � � . �

� � � � � � � � � � . � % . � � � � � � �

� � � � � � � � � . � � � � � � � . � � � � � . � � � � � � � � � � � � �

� . � � � � � . � � � � � � � � . � � . � . �

� � � . � . % . � � � . � � � � � � % � � � � � � � � � � . � � � % % � � � � �

� . � � � � � � � . � � � � � . �

Page 137: Goldrei.classic Set Theory for Guided Independent Study

+ . / + 3 + 4 + 3 . 4 9 . 4 : 3 9 4 9 + J + : 3

. : + 4 : M / . / + : +

: / + J M 3 : 3 4 . M 9 + J . M 3 / 9 3 4 + . + 3 : + . 3 / : 3

3 4 / . 4 + J : / M ~ : + : + + . / . J + + : + . M . : + ~ +

4 9 / M + . ~ 4 M 4 . 4 M 4 9 + M ~ ~ M : /

+ 3 . 9 4 : 9 : 4 + + : . M / M / +

/ : M

/ + J : 4 +

. 3 4 ~ + . / . 4 J + 4 + + 4 . 4 + 3 : + M 3 : 4 ~ . ~ / .

M

: : / : / M 4 4 . .

+ 4 + / J : 4 +

+ 4 3 3 + 4 + + + / : 4 : M / 3 4 M M . J + 3 : + M . + 3 4 3

M J + M 4 + M ~ M / . 3 M 3 + : / M 4 +

M ~ ~ : / 9 M ~ 3

. + 9 + + + + . / + / + . M / + M / + / 4 : M J M 4

~ + ~ + M 4 4 M M J 4 M + + :

: M / M + M 4 M M + : 4 J + : J + 9 M : / M 4 4

9 9 + + . J + M / + . / 4 : 4 : / M + + 4 4

+ ~ ~ J : 4 + 3 M + + M 3 M + 4 + + : 3 / M

: + : /

Page 138: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � " � � � � � " � � * � � � " � � �

� � � � � � * � � � � � " � � � � C �

� � C � � � � * � � � " � � � � � � � � � " * � � *

� � �

" � � � " * C � * � � " � � � � � � � � � � � � � � � �

� � � � � � � � � � * � � � � � � � � � � C � * � � �

� � � * � � � " * " C � * � � � � � � � � * � � � *

� � � * � � � � � � � � � � � � � " �

* " * � � � * " � � � � � C � � � � � � � " � � � � � *

� � * � � C

C � " C � � � � � � � C � � � "

� � � C � � � � � � � " � � � * � C � � � � � � "

� * � � " � � " � � * � � �

� * � � � � � � * � � " � � � � � � � � � � " �

� " � � � � � � � � � " � � � � � � " " C

� * � � � * � � * � � * � � � � � � � � � � �

� � � � � � � � � � � * � � * � � � � " " C � � * �

" � � � � � � � � � � � * � � � � � � C � � � � � � � � "

� � � � � � � " � � � � " � � � � � � " � � � * � �

� � � � � * " � � � � " � � � � � � �

� � � � * " � � � � � � � " * � � � � � � C C � � � "

" � � � � C " * * � � " � � � � � � � � " " �

* � C " * � � � � � C � � * � � C � �

� � � * � � � � � � C � � � " * � � � � * " � � � � " � �

� � � � � � �

" � � � � � C � � "

� � � * � � � � � � � " " � " � �

� � � * � " * C � � � � � � � � �

Page 139: Goldrei.classic Set Theory for Guided Independent Study

� � $ � � % & ' � ) � & � ' ' & 0 & ) & ' � ) = $ ' 0 ) �

) %

0 G $ = & $ ' 0 � � = � � G $ � = ) � $ � 0 G � � ' G ' = � $ � $ '

= � ' 0 ' � � ) & � & 0 � � � % & 0 � 0 & 0 � � % � & � ' % '

) G � 0 � ' � � � � � � & ' ) % ' G � � $ ' $ � & $ � $ 0 ) = = '

0 � � & 0 � % � � & � ' � ' � 0 ) � � � % 0 ' ' G % % & $ � & $ � ' % ' &

' ) � % ) = ' & 0 � G � $ = & $ 0 0 ' G � � G � � � ' % � � $ � & $ � 0 & = � ) $ & '

� � & % ) G = G ' G � % & � $ � ) % � & 0 ) ' � G $ $ � ' � &

$ % ' ) ) $ 0 � ) $ � & ) & ' � $ & � $ � ) % & ) ) = = � % % & � $ � & ) �

& 0 � $ $ � � 0 & � � G & 0 � % $ � � � $ = � ) G ) � % � � & ' % � $ & � & �

� % % & $ � $ � & 0 ) % ' � % � % 0 � 0 � $ ) = = $ & ) & � % 0 � � & � ' � G 0 �

% �

0 ' & 0 & ' ) % $ � & $ � & 0 � %

� & ) % � % & G ) = % � G $ % 0 ' 0 ) &

� 0 �

� 0 � %

� & 0 ' & 0 &

' � 0 ) & 0 ' & 0 & $ � % % � �

� & ) % � % � $ � 0 ' 0 ) ' G % & � ) � % � %

0 � ) � 0 ) � � ) G � � G � ) % & � $ � &

� � � 0 0 � ) % ' & 0 � ' = ' � $ & ) & � � % 0 ' =

' $ � ) = � G % & � � % � � ' ' = � $ & 0 % ' G � � % � G ' $ 0 � 0 � = =

� � $ � �

� & 0 � � $ % ' & ' & � ' %

$ � & 0 � � 0 ) & � ) & = � %

Page 140: Goldrei.classic Set Theory for Guided Independent Study

1 2 3 4 6 4 8 ; 4 2 8 > 1 > 4 4 B ; 4 2 > 8 1 4 8 B 2 4 > 6 3 4 ; 2 ;

B 1 3 > 2 3 4 6 4 _ B 8 > 1 ; 3 8 ;

B 1 3 > 6 ; > _ 2 > 1 3 > 4 B 2 B 1 >

3 3 2 > _ > 4 4 1 8 2 8 > 1 3 B 2 B 1 4 B 8 2 B 2 B 6 4 1

> 2 B 8 8 > 1 > _ 4 ; 4 2 > ; 3 > 2 B 2 B 1 2 6 4 _

2 3 4 ; _ B ; 2 4 _ 2 > 2 > > _ _ > 8 4 B 2 4 8 2 > 2 B 1 > _ 8 B 2 4

B 6 4 2 B 4 B _ B 6 4 > 6 8 8 8 > 1 ; > 2 B 1 6

; 8 4 ; > ; 4 2 1 2 3 ; ; 4 2 8 > 4 ; 3 _ _ 8 1 4 ; 8 B 2 4 2 3 4 2 B 1 B 4 2 4 1 2 2 >

3 3 2 3 > 4 ; B 4 1

4 2 ; _ > > 6 > _ _ B 2 3 4 ; > 6 6 > 2 8 4 ; 4 > _ B 1 2 2 > > ; ; 4 ; ;

2 > ; 2 8 3 B 8 B 2 6 4 ; 2 3 4 8 4 B > 3 B ; 2 3 4 4 8 4 B ; ; _ 6 4

8 8 4 4 4 2 B 4 B 2 8 B _ 4 1 8 2 8 1 > 2 3 4 4 B ; 8 ; 3 B 2 8 2 8

B 4 8 4 4 6 4 _ B 2 1

3 > B 2 8 ; B 1 4 _ 4 4 6 4 _ B 2 8 > 1 4 2 4 4 1 4 2 ; 4 2 B 8 8 ;

6 4 4 8 4 > 6 _ _ ; 4 2

4 6 8 > 6 B _ _ ; 4 2 B 8 2 3 4 1

6 B 1 8 2 4 > _ _ 4 2 B 1 8 1 2 4 1

3 B 8 1 4 8 _ 4 1 4 6 4 _ 2 1 8 > > ; 4 _ _ 4 _ _ 4 >

4 3 2 4 6 6 1 8 8 _ B 6 8 4 > 1 ; 8 > 1 > 1 4 8 1 4 > _

4 8 1 4 6 > ; ; 4 2 8 1 2 4 > _ _ > 8 1 4 4 4

> ; 4 2 B B 1 3 > 3 B 2

; 2 4 6 4 B 6 4 8 4 8 > 1 B 1 3 4 1

4 4 2 > 1

2 8 4 > ; > 2 3 B B 8 4 2 8 > 1 > 2 B 2

> ; 4 2 B 2 B 1 2 3 B 2 8 B 1 ; 4 2 8 4 B 1 4 B _ 4 2 > ; > 2 B 2 8 2

; 1 > _ ; 2 3 4 4 2 3 B 4 ; 2 > 1 8 8 > 1 > 1 8 3

> _ B 6 1 2 4 4 2 3 2

4 6 1 8 2 > 6 > _ 4 8 > 4 6 _ 8 2 4 2 4 6 > 6 > 6 8 1 2 1 4 8

B 1 2 4 1 B 1

3 8 3 B 6 4 > 2 4 4 6 >

2 8 8 ; 8 ; > 8 1 2 > > 2 B 1 2 4 6 > 4 8 B 4 B 8 ; B

Page 141: Goldrei.classic Set Theory for Guided Independent Study

" # # " , " , - . - " # 0 " 0 . " 0 5 " , # 5 5

,

- . " # 5 0 L # # # # " . " , , - # , 5 " L # # " 0

. 5

5 0 5 " L , . - # " # - 0 # " . " . , # - . , L - # - . , 5

L # # , a a 0 " # # " , , , 5 . L # , " , " , # 5 0 , 0

, a # L 5 L 5 , 5 . 5 # L # 0 5 " # 5 , " 0 , 5 #

. # - " - 5 # a a - a . L , 5 0 - , # 5 " 0 . " " 0 L

- . 5 . a " L 5 . " - " " , # a a - - a . 5 " # " - . L " ,

. - , 0 . # , 5 " 0 L , # L 5 " - " , 5 # 5 # . #

. L a L # , 5 # # " 0 - , L a - . # " 0 . L , " 0 " , # . L , 5 . a -

# L # 5 - - # #

# " 0 # a , 5 - " . " 5 " - 0 , , " , 5 . 5 . L # L , . " , " , # 5 , 5

.

0 L

0 L

.

a " 0 # 0 " 0 , - , - a L . , , # 5 , " , - # 0 . L , 5 , 5 , 5 , 5 . a a

" , 0 # # - " , " 0 . " # 5 - # 5 L , # # L -

# . a a , - . " . a a a a 5 . - . 5 " , # L , - , 5 # a -

5 # " 5 " 0 . , 5 L # " 0 # L L , L # 5 - L " , # 5

. L - # " 0 " " L . L 5 , , " . 5

0 " " 0 L # # , # 5 L , - 0 . 5 , a 5 L

#

" # 5 # 5 L # a . L . - # a # - , .

0 5 0 # L . , L # 5 - " L

, # 5 " 0 5 . . 5 " - " 0 . " 5 # " 0 "

. L

. - a - " - 0 # 0 . , - # 5 # L " 0 , - , - . L " , a . L a - L . , 0 # L

. , - " , 5 L #

, 5

Page 142: Goldrei.classic Set Theory for Guided Independent Study

� # # & � - � & - � & -

& - J # L J L � � # L � & -

& L � #

# # - #

& - # L L � g � # � & - & - # L L J L

L & g J & L J L � g & - & w � � & - � g # - # L # & L & -

w & & w g � # w & - # # L g L & J & & w � -

� g # � # - #

& L � # - � g w & � L J # J w # g L w L & J g # # � #

# w & � & L # - � # & # & - - - � �

� � L � - � � � � L � & - - L � & -

w - w � L � & & &

L w � & # # & � & �

# w # � - & w � � & � &

g � # & � # & &

� # & � & &

� # & � & &

& J � # & � � & # L # & # w L � & - # # & � #

& & � # & g L & � # � - � # # L # � # � # - L -

# � # # - L g & # g # # L # w L - & � # - L &

L � # - � g � L & # w # # # & �

& � # J # � # & � - - & # w & # �

� & � - L # # � - # w g L -

� J # # w � J # & # w � L � L &

# � g # L # � � � - # L � - # w & # L & & - g & w

� - - # # - L � � L L # � L � � L � & - � &

L L � # & - # L - & w � L J � & w � � � g # � - � - & � & &

g J & � L - & � � - & L # � w # L � & w - # w

# � - # L � L # � # # - g L � g & # L # - - & � # w

w & # w � # g � - L & L J # w # � w # # - # w � g - & # - L � - & � & &

� � # L # � J L & # w # g � - � � L � # g � - L � g w � L # L �

# L g - & � & L - J & � � � L L # L � # & &

g # g & g # # � � # # - L & � � & - � �

L # g J # g & � � # w & # � & �

- # # & # L & � � & � & L � #

� �& # L

& w � - -

# L L #

Page 143: Goldrei.classic Set Theory for Guided Independent Study

. 0 0 5 6 8 8 6 9 : 8 : ? 8 : ? 6 @ : 5 @ . 6 : ? 8 : ? @ 6 8 : 5 . 0 :

0 : I @ . 0 6 8 : 5 ? 8 . . ? 8 8 5 6 : 8 6 : ? 6 8 ? 8 : ? 6 @ : 5

8 0 9 0 : 8 @ 6 : 5 @ ? 6 8 5 8 6 6 : 8 ? 8 : ? 6 @ : 5 9 8 I . 0 : I

? : : @ @ 6 0 . 6 0 . @ 6 6 8 8 8 : 5 : 0 : : 5 6 @ 8 ? 6 :

@ : . 6 @ 6 . 6 @ 8 . 8 . @ 5 0 6 8 @ 0 I I 8 . 8 . : 5 8 0 . @ 6 : : 0 ? :

8 . 8 . : . @ 8 . 5 ? 8 6 8 0 . ? 8 8 6 9 : . 6 : @ ? 8 8 @ 6 @ 8 . @ 5 0

8 . : 8 . : . 6 @ 8 . ? 8 8 @ @ . 6 : 6 0 . 5 @ @ @ 8

8 0 ? 8 : 5 8 : I @ . 0 6 @ 8 . : 8 : ? 9 @ 8 5 . 6 9 8 I 6 : 0 .

5 : 6 8 6 @ : . : : 0 8 . : : 0 I : 6 5 0 . @

0 . 8 ? @ . : : 0 I : 0 5 . 0 I 5 8

. : 0 I 5 : 6 0 . 8 @ 5 8 6 0 8

8 : 5 @ 5 8 I I 8 8 ? @ . 0 . : . 8 ? .

0 @ ? 8 : @ : ? 5 6 . : I : ? 8 @ : 0 8 . 6 : : I :

8 @ . 8 . : . @ : . 6 8 6 @ . 8 6 : 0 @ I @ 0 8 I 8 0 0 ? 6 0 I 8 ? :

6 @ 9 9 @ . @ 9 6 : 5 8 I . 8 6 9 0 : : : : 0 . 6 5 : 6 8 : I @ :

6 8 8 ? @ . 6 @ 6 @ 8 . 8 9 5 5 @ : I : 5 5 6 0 . 8 ? : 0 I 6 8 6 9 : . .

9 8 I : . 0 @ I 6 9 0 6 0 . 9 0 6 9 : 0 : @ : 8 0 6 : 0 : : 0 I 9 0 6

6 9 : 8 : 6 0 @ I 9 5 6 9 : 8 . 6 : . 8 6 9 : 8 I I 8 @ .

0 . 6 9 .

9 : 8 8 8 6 @ 5 @ I ? ? : I 6 @ 5 @ : . 6 I 0 8 0 I I : . :

5 : : ? I ? : 0 0 6 9 : 0 @ @ 0 . 5 8 @ : 6 5 0 . 0 : : 5 0 I I I : 0 : 6 9 : ? 8 8

6 8 6 9 : : . 8 6 9 @ 5 5 : 6 @ 9 : 9 : 8 : @ : 5 0 . I : ? . 0 @ : 5 ? 6 :

5 9 8 0 6 @ : . 6 . @ : 8 . ? 8 8 . : 5 : 6 8 6 : 8 : ? :

0 . 0 I 0 ? 8 . 0 @ ? 8 . : 8 . : . 6 @ 8 5 8 6 8 . ? 8

8 0 . 5 : 6 @ 5 6 : 8 :

6 5 8 9 . 8 6 @ . 9 : ? : 6 6 . 8 5 6 8 8 0 @ . I . ? : . 0 6

0 ? ? : 0 9 5 6 @ 0 8 . I @

0 : : : . : . @ : 6 @ 6 8 6 5

0 ? @ . 5 @ 6 : 8 : 5 0 . 6 @ 5 : 6 @ @ . 6 ? 8 5 . @ 0

0 ? 0 . 0 ? 0 : . 0 ?

9 0 . 5 6 8 9 : 9 8 : ? 6 : @ . 6 9 : 8 : : 0 . : 0 ? 0 5 : 5 5 : . I I

I : 5 5 6 9 . 8 ? : 0 I 8 9 : ? : 6 9 : ? ? : 8 . . : I 6 8 @ ? 0 6 : ? 6 .

. 6 : . : 6 : : ? @ : 8 0 : 0 6 8 9 0 6 0 . @ . : : 9 0 : 6 :

5 8 ? 6 8 ? : I 0 6 @ 8 . 5 9 @ 8 : : 6 : : : . 0 .

5 @ . @ : 6 8 . 5 0 . 8 8 : . 6 @ 8 . 5 : ? 0 ? 8 @ 0 8 6 0 6

0 @ 0 . : .

@ 0 . : .

Page 144: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � # % & � % � # � � � & � � % # � � 4 4 � �

� % 4 4 � & % � 4 % � � � � � � � � % 4 � � � � � � # 4 � � � & � � � � � # � & % � � # �

# � � % # � � # 4 # � � � � 4 � � � � � � � # � % �

# � # � � � � # �

% � � � � # � � � & # # �

% � � � � # � � # � � � � # � � � �

� � 4 4 � � � % � % � # � �

� � % � % � � � � � # # � � � # # �

# � � �

� � � �

# �

� � � � � � 4 4 � 4 % � � � � # % � � � � � % # � % � � % % & �

% & � � % # � � # � � 4 � �

4 4 � � 4 # # % � # � # � # � & � � � # � � � � #

% � � � � � � � 4 % & � # � � � � # � # � � 4

# & � & � � & # � � � � % # � � � # # � � � � # 4 � #

� # � � % � � % � � � 4 # & � & � � # � � � � # � � � &

� # % � �

% & � � � # & 4 4 � �

� & % � � � � # & 4 4 � % � % � � �

% � � & � � � � � # % � # � � � � � # #

# � � % � % � � � � � # �

# � & % � � � � � � # � % �

# # � 4 � � � % � � % � # # � # � % �

% � � & � # & � � % # � & % � � # � � � � �

� � % � � % � � � � & � 4 � # � & � � � 4 # % �

� � � & # � � & � � � � � � � � # & � � � � % � � �

# � � & % � � � � � # & � � % 4 % � � � � � % � � # � %

� � & �

Page 145: Goldrei.classic Set Theory for Guided Independent Study

� ! $ & ' ( ( * ! + ( $ * / ( 1 $ * / / * & / ( 1 * ! ( 1 + & ' & (

/ ( & 1 * ! ( * ! ( & ( / ( & 1 / * & ( & $ & ( 1

* ! + / $ ! & ( / ' & ( / * $ ( / / & ' ' ( * ! & & & ! ( ( / * ( ! + 1 &

* & $ + + * / $ ' & & * ' ( & 1 / 1 & * ! ! & / / * ( !

* ( 1 & $ ' ( & ( (

* (

& !

� & ( * ! * + � / $ * ! ' * & / * * ' ' ! ( & & 1 * ! $ '

* & & & ! ( * ' ' $ ' ' 1 & 1 & & ( !

! & ( ! � $ & 1 ( ' $ ' 1 * ' ' *

( ' & * / ! * ' $ / ! ( ' * ! ' *

& $ & 1 * ! * * ( * ! ( 1 * * *

( & ! ' * / ( ( & / * * ' ( 1 ( 1 (

& ( 1 ' ( ! / ' * 1 1

' ( ( * ! / & 1 ( & 1 * & ( * ( (

! $ 1 $ 1 1 + * * ( * ( '

& & 1 & + &

( 1 ' &

! ( ! / $ / ' * ! ' &

( / ( * & * ! ' ' $ / ! ( ' (

! ( ' ' & ( 1 * ' * ' ' ( ( *

' $ / & 1

( & ! / � & * ' * ( ( ( *

$ ! 1 & 1 ! ! *

( 1 * ( ! & ! ( ( ! ( /

& * 1 & * & / * ( ! ( ( * ' 1 * ' ' !

/ � $ & � $ & ( $ � ( ( ! / $ * ' * *

* 1 + & & & / ( 1 ' ' ( * !

&

! * ! / ( / / 1 ( * *

& / & * & / ( ( !

( '

Page 146: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � " " % ' '

" � � Q � R � � � Q � � � � Q � " % " �

� " � " � � � � R � R � � R � � " � � Q � R R � Q � � � Q � % R � Q �

� R � � Q � � � Q � � Q � � " � � � � R " � " � "

R % � " �

" � � � % R � �

� � � � � Q ' � " % Q ' ' R R R � � " " � � R �

Q % � ' Q � Q Q ' � R � R � � �

� � � Q ' ' � " � Q � R � R R " � " � � R R " � " � �

" � � Q " � � " " % � " " % � � �

Q � " � Q � � � R

� � � Q � Q � " � � " � � R " % �

R � � R � � R � � � % � " % R R � � Q R " � " � � � ' "

� � � � " � " ' � R � � � � " � � " � Q R R �

Q � � � " � � � % � � " � " � � Q � � �

� R ' �

" ' � ' " � �

� � Q % � � R

" � � % Q R � � � � R � " R � � � R � � � R � � � Q ' " Q �

" ' � � R � � � � % " " Q � R " � " � �

" " % � � � " � � " % " � � " R � � Q � " % � � � % � R � " �

R � � � % � R � " % " ' � R Q � � " %

Q � R � " � � Q � � R R � " ' " "

Q � R � � � R � � � " � � Q � " Q

� R � � ' ' ' Q � % R % " � Q � � � � R " � " R �

R � � � Q � R R � � � R � Q � �

� � R � " � � " " � � R � Q R � R " Q � R R � Q � % R % �

" ' � � � � � � R ' " � � % " " " � Q � � Q �

� � � � " Q ' '

Page 147: Goldrei.classic Set Theory for Guided Independent Study

0 1 2 3 5 7 5 : 0 ; < 2 0 1 < < : E 5 7 3 5

< 2 5 : : E E < : 5 1 < E 2 E

1 5 < 7 7

: 0 1 E ; 5 < 5 E 5 0 1 E 1 ; E ; E ; : 3 0 E 2 : 0 E 5 ; E 0 < E :

7 E 0 < : E 1 E 5 : 5 : E : 0 2 5 : 3 E : E <

E 3 ; 2 E 0 1 E ; 7 E 0 1 E 5 ; ; E 3 5 : 2 0 5 : 5 : 3 0 ; E 2 0 1 E ; 5 5

1 E : E 5 3 E 2 5 : 3 E 3 1 < 7 7 3 E E 5 E 2 ; E 3 3 E < 7 2 < 2 5 3 5 1 E

1 ; E ; E ; 0 E 2 : 0 1 E 5 ; E

1 5 1 < 0 0 1 E :

1 5 < 0 2 1 E :

< : 5 : E E : E < E 2 7 E 5 ; < 2 5 5 3 0 ; < 2 : 5 : 3 2 E 5 3 E 0 3

3 1 E ; E 3 E 1 0 1 < 5 ; < 7 7 E 2 E 3 E 1 E ; < E 3 1

1 < 0 ; < 7 3 E 0

; 5 E 1 < 1 E ; 2 2 7 E 5 1 5 0 5 1 5 7 3 5 ; 3 E 0 2 E 2 < : < E

: 2 0 E E 0 1 E : 5 ;

5 E 1 < < ; E 3 E E 1 0 1 < 0 E < 0 0 5 E 7 E E : 0 3 1 5 0 1 < 0

1 5 0 1 2 < 1 E :

E 0 3 : 5 7 5 5 5 ; E 7 5 3 E 7 < 0 2 : : 2 E E < : 1 5 1 E < ; E ; E 7 < 0 E 0 5 E < 1

5 0 1 E ; 3 2 : E E : 5 0 1 < 0 2 3 2 : : E 3 5 ; E 3 < 7 < : 2 E ; 3 E

3 < 7 3 5 2 : 2 0 E E ; 2 0 1 3 : 0 1 E 5 7 7 5 2 : E E ; 2 3 E

Page 148: Goldrei.classic Set Theory for Guided Independent Study

( ) * + , + , ( + , 4 ) 6 6 * 4 6 + ? ) * + ? +

? * H 4 * ? ( 6 + H 4 + ( 6

) + , * ) * ( Q 6 + * * H ) 4 4 6 * * 6 6 4 + * ) ( + ? +

* , ? * Q H * ? ) * * 6 + , * H ) + ( ? 6 ( H ) H H 4 * ) 6 * Q + , ? * * 6

, ( , 4 ) ? H ( * H ) ) * + ( ( * ? * + ( ? * ( H 4 6 + 4 ( 6 ? (

( H 4 6 + * ) * + ) * ) Q ( ( ) + ? 6 4 + ) ) ) * )

* ? * 4 ( 6 + ( ) , ( , , * H * 4 * , * ) * +

( * * 6 6 + * * H )

* 6

+ 4 ) * ? ) + ( ) , ( , ? + 4 ) ? 4 * + 4 ( 6

+ ) * ? ) + ( ) , ( ) ? H , ? + ( , ) , * ) *

) * ? + * ( 4 H ( * H ) ) * + ) , * * ) ( 6 ? 4 ) * ) ( , ? ? H * , *

( ) 4 Q * ) 4 * ) ( ) ) * + ) ( Q * ) ( * ( , * ) ) * ) ? H * 4 * *

+ , * H * ) ( * 6 + H Q 6 * H ) , + , ? + + ) + , * H * ) ( * ) )

( , , * + , * H 4 * ( * 6 4 ( 6 6 4 + * 6 ( H * 4 6 * H ( ,

, * 4 ) + , * 6 ) ( * , ? * ? Q Q * H * H * * ( 4 6

+ , ? 6 ( ( ( + ( ( H 4 * H 6 ( , * ) + ) ? ) , * 6 * + , * ( H *

) , ( )

( * , ? + , * 6 * + , * 4 ) 4 + * ( H

( H ( ) , ( ( * 6 * * ( * + ? 4 * ( 6 * * * ? * , * ( H ?

? Q * H + H + H * ( * ) , ? * , * ? 6 * Q ( 4 ( 6 *

? * ( 6 H * ) * + ) ) H * + , * H * H 4 ( , * 6 * H ) 6

, * ? , ? H * , * * * H ( 6 + ( ? Q 4 ) 4 ? ) * 6 ( * H

) * H ) * 6 * * * + , * ( ) 6 ? H ? ( * +

( ) + ( * 6 * 6 * ) ? H ( + (

H + * * H (

* ( 6 * * H (

, H * * H (

? 6 ) ( (

* 6 ) + * ( 6 + 4 + ( ( * Q * * ) ( * * ( *

) 6 , * * ( * Q Q ( * ( H ) + * * H * ( * )

) * ( * * H * ( * 4 6 ) ( ( ( + Q ( ( ? ) + , ( ,

Page 149: Goldrei.classic Set Theory for Guided Independent Study

" $ & (

4 ( ( &

> 4 ? 4 ? @ A @ B > & A @ > & ( 4 4 " ( 4 & $ 4 ( " A 4 ( ? B @ ( & B > @ B

4 ( @ ( &

( ? @ ( @ ( &

4 (

( @ ( &

4 @ ( &

4 (

@ ( 4 ( B 4 & $

4

4 $ $ B > 4 ? & ( 4 B " ( " " $ @ ( > ( " A " ( " @ @ & @ B 4 (

& & 4 ? @ $ @ ? " B & ( B $ A " $ & > & ( ( ( B > & ( 4 & $ & & ( B " @

? & B " & $ $ " A & A & & B > B 4 B > & ? ? & ? " @ (

@ ( & & & A & & B & " " A ? & 4 ? ( 4 B & ? A & $ B > & A &

" & " A > 4 4 ( B 4 B 4 & $ 4 ? ? @ $ " B > & & $ &

& ( B ? " " A & ( 4 B 4 " ( " ? B ? & > " " & 4 B > B > & @ ? & B > @ B 4 ?

( 4 B & $ > " > " ( & " $ ? A A & B > & @ A & ( B & ( 4 ( 4 ( " @ ?

" ( & " A B > & @ ? & B > @ B 4 ? ( 4 & @ ( " ( & " A > & ( 4 4 ( B B > & A & 4 ? " &

4 ( B B A @ 4 & @ " B B > & " $ $ " 4 ( " ( ? B A B 4 " ( 4 > " & A ? " B @ ? & ?

& B A 4 4 ? B " 4 ( B " & ( & ? & B ( " B ( B " & B & 4 @ & " " $

@ ( & & & B > & ( " A & ( 4 B 4 " ( " & " & ?

4 @ ( & 4 @ ( &

" B > & 4 ? &

( $ 4 & " & @ A $ 4 & A & ( B @ B 4 & & ( 4 B 4 " ( > 4 ? 4 ? & $ $ & ( & > @ & & A ? ? & B

4 " & ( " > @ & B " @ & ? & ( 4 ( @ ( & @ ( > & ( 4 B 4 ? (

4 ( 4 B 4 & $ > & ( 4 ? ( 4 B & @ ( > & ( 4 4 4 ( ( 4 B &

@ ( &

( B 4 B 4 & $ 4 B & ( " A @ $ $ @ ( B > & A B ? > > @ B

4 B ? 4 ? B > & 4 & " " & ( & B > 4 " A @ $ ( " & B > @ B

4 ? ( " ( & B @ ? @ ( & @ ( 4 @ ? ? & B " ? " > @ ? @ $ & @ ? B & $ & & (

$ & B & B > ? $ & @ ? B & $ & & ( B & $ @ 4 B @ B ? @ 4 & B 4 " A " B " ? "

B > @ B 4 ? ( 4 &

4 ? " ( B " ? " $ " ? B > & 4 ( 4 @ $ 4 B " & > @ & B @ B " A @ $ $

" B > @ B & > ? > ? 4 ( > ? @ ( & ( & ( 4 B 4 " (

" ? " > @ @ ( & B B > & ( @ ( & ? "

B > @ 4 ? " ( B "

& & B @ 4 $ ? " ? > " 4 ( B > B 4 ? " ( & " ( & A & $ & B " " @ ? @ B " B > & ( & B

& & A 4 & " B & A B > & " & ( B B > @ " A ? " $ 4 " ( B " B > & & & A 4 ? & 4 $ $ @ B @ $

" " A & > B > 4 ? " ( " ( & B 4 $ $ ? > " B > @ B ? B A 4 B $ 4 ( & @ ? 4 (

& " A & & A ? " B > & (

Page 150: Goldrei.classic Set Theory for Guided Independent Study

& ' )

' / 0 1 & 1 ) ) / & ) : ; & < < 1 : > > & ' ) ) 1 & )

& ; ) ' > 1 > / > 1 / : 0 ' : : ' : ' ) 0 < < 1 : > / & > 0 : / : ' ' )

& ' 1 > 0 > < 0 ' ) & 1 0 ' > ; ) & 0 ' 1 : / : > & 0 : ' : > 0 < ) < > : & )

> / ) : < < : 0 ' < ) &

: ; & < <

/ < < 1 > ) & > & > > 0 0 ' ; '

; : ) ) 0 ' :

: ; ) / & ) > > 0 1 1 : ) ) ) ; : 1 : > & >

: ) > / & > : 1 : ) ) 0 1 > ; 0 < 0 ' ; ) & 1 0

: > & >

1 0 ' > ) 0 ' > ) / : / ) 0 1

/ ) ) < > : > ) < & : < < : 0 > 0 '

) 0 < < ' : 1 / : 0 : ' > : 0 ' : & 0 : ' : > >

0 1 ' : > : ' > : / ' > ; ) : ) 0 > ) 0

) 1 > & > ) > ) & < ) : ) < & : :

0 & ' ) : & > : ' ) : )

& ' ) ' ) ' ' ) : > & ' ) : )

) ' > 0 : ' 1 : > & > ' ) 0 0 ' > & >

> 0 ' > ) > : 0 ' ) < 0 > 0 ) > : ) > / ) ; ) > > / ) ' & ' )

: ' > ; & 0 0 ' 1 1 > 0 : ' & ' )

/ 1 0 : ' > : & ' ) ' 0 & > 0 : ' : > /

; : ) > / ) < & 0 0 ' > : : : > 0 0 < : 0

& / ) < > : ) ) > & > ; ' ; < 0

: ' < 0 & ' : ' < 0 0 ; : ) ; > :

& ) 0 ' & > > : > < : : ) ' ) & < 0 > : & > & 1

0 ' ) ' & ' ) ' 1 : >

& ) & ' )

: & ' ) 1 : > ' 0 0 :

0 ' & ' ) : > < : 0

1 : / & > ' ) 0 < ) ' 0 > 0 : '

& ' ' > ) & < '

1 ) 0 1 0 : ) 0 ' 0 ) : 0 ' ) : >

) & > 0 0 0 ' ) : 0 /

Page 151: Goldrei.classic Set Theory for Guided Independent Study

# % & ( # % ( * # ( / 2 ( & # 5 2 # / & * 7 ( / ( * ( 2 # / * &

# & @ * 5 2 # / & 7 ( # * # & # ( # & ( % * # 2 2 ( & 2 7 5 (

@ 5 @ * ( 7 ( * 7 5 / 5 & ( ( # % ( ( 5 / ( 2 & 5 * ( & @ ( ( 7 ( 2 % ( 5 ( @ ( *

5 / @ @ ( @ & 2 @ # @ * & * 2 ( & 5 # / & * 7 5 / 5 & ( 5 & # 7 (

/ # / 2 * 5 ( # / % # # # % 5 ( 2 * ( # 7 * # ( 7 # 5

@ ( @ % # ( ( % / 2 ( / & @ ( # % ( # / 2 % & / & # # / # / ( / 5 # / 2 # / (

% # & # * / # / ( / & @ ( % ( ( 2 ( 5 ( & 5 # / 5 2 2 # ( 5 ( 2 ( 2 5 ( % # # 5 2

& @ * / & # / 5 ( & 5 # / / 5 & # & @ ( & @ # 2 ( * / # # % 2 ( % ( 7 * (

* # @ ( % / # / # / & 7 5 / 5 & ( ( & / & ( / ( ( 5 2 ( ( 5 ( # #

2 * % & @ ( % ( & @ #

# 2 ( * & @ ( / & 5 # / 5 2 # / & # / & * & % # ( & *

5 2 # / & * ( * / # * # 5 2 5 / / # # ( 2 ( & 5 7 7 # # % 2 (

( # / * 5 / / 5 ( & * 2 @ 5 2 5 2 @ * # & 7 5 & ( ( 2 2 & # * % & #

( 2 # ( @ * & 2 # 5 & @ # / ( 2 & ( % 5 / # 7 #

( & & ( ( / & # ( % 5 2 5 2 & * & 5 # / ( / 7 5 & ( ( ( % 2 # 2 (

2 @ ( % ( ( ( ( % # * ( * 2 ( * 2 # / ( 2 * /

# % 2 # ( & # 2 2 7 * ( * 2 2 ( ( & 5 2 ( / 5 2 # & * 7 (

( 2 / # 2 ( ( * # ( 2 ( ( & @ # # % # ( # / ( # % 2 # / #

% % 2 5 / % ( 2 7 & 2 & @ & & @ ( 2 ( # / # / / ( 5 ( % 5 # / * 7 5 2 # / & * 7 ( &

@ 7 ( ( 2 / ( ( 2 2 * % & # 2 * & @ * & & @ ( % ( 7 & * & 7 ( * & & @ ( % 2 & 5 ( # / ( ( (

& 2 7 2 % % 5 2 5 ( % ( 2 ( ( & # ( 2 7 # ( % & 5 # / * 7 2 & @ * / / * & * 7

/ ( 2 2 ( @ * ( ( & ( / ( * / # / & # / & 5 # / % #

# / & # # @ @ ( % ( 7 & # @ ( % ( # 2 ( % 2 ( 5 2 # / & * 7 (

5 & 7 7 @ ( # 2 5 & 5 ( % 5 # / * 7 2 5 / * / * % % * 2 # # 2

2 # & * & & ( * 5 # / * 7 ( * % 2 2 & @ ( ( / & % 5 / & @ ( & % # * / & # 7 /

Page 152: Goldrei.classic Set Theory for Guided Independent Study

� � � ! � � $ % ! ! % ( ! � + , % � $ � � 0 $ � ! , $ ! % , ! � $ $ + ,

( $ ( + $ % , % � , 0 ( % ( � 0 0 � (

, + � � + , ! � $ ( ( + � , � $ % ( + � $ � $ % ( � $ ,

� % $ % � $ $ , ! % $ % ! ( % ( � $ �

� $

+ ( � , � + , � $ ( ( $ , $ % � + � , ! � � $ ( � � ( � + $ + , � $ $ (

� � � $ ( $ � � ( + + $ ! % � + � , % 0 % , ( $ � � , $ � + , % , � + � ,

� � 0 $ % ! 0 $ $ ! � , , $ % � + $ ! � + � , % 0 + ( � ! $

+ , % � , � ( + � + $ � + � , % 0 + � $ + % $ + , , + � $ 0 % , � ! + , � % , $

% $ % ! ( + , � $ % ! ! % ( $ � � ( � , � � % , � $ ! $ ( 0 � $ + ( $

� � � 0 $ �

% 0 � $ , % � $ $ + � $ , $ + , % + ! $ % % � , � , $ � , ! �

� � $ , � ( $ $ ! � $ ! $ ( $ , � $ � ! $ � �

$ $ � $ ! $ ( 0 , % � ! � + $ � ! + + ( � � $ 0 � + � � $ , % $ , % 0 $ � ! $

� % ! $ � + % � 0 0 � ( $ , $ % , � + � ,

+

+ % ,

$ � , + + , + � ! $ , $ $ $ � + $ % $ 0 0 $ ,

% $ � ! , + � $ ( � � $ � % � + ( � , $ � , $ % ! $ + ! $

� � � $ � ( � + $ � � ( + , % % % ! $ $ + , $ ! � (

, % $ $ + , + $ + � , ! � � , $ � � � $ � � $ $ 0 0 � % , $ � , $

+ 0

� % $ $ ( � , � � $ � , % 0 $ + � ( ! $ 0 � + $ 0 $ ( � � � � �

( � , � 0 $ + ! ( ( � % � � $ ( $ � , $ � + $ ! % � + � , % 0 ( + ( � � % 0 $

$ , ! � $ � $ � 0 0 � + , � $ � $

Page 153: Goldrei.classic Set Theory for Guided Independent Study

) - ) 2 - 4 5 ) ) 9 ) - 9 2 - 5 ? )

? E - 4 G 9 G ) 2 J ) 5 - G ) - E ) G 4 E 9 2 - 5 ? )

G 2 G 9 2 - ) 2 5 ) 5 2 E ) 2 ) 4 2 9

2 ) G ) 2 J ) E - ) 5 ? 9 G ) - E 2 - 2 4 G ) E 9 2 E - 4 9 )

5 - E 4 E 9 ) - 2 G 4 2 J 2 ) 4 ) J ) ? 4 2 J 5 - E 2 E - - 2 G )

5 ) G ) ) 2 - J ) E - 4 )

J 2 ) G ) - E 2 - 2 - E ) - 2 - 5 ? ) 9 ) 9 9 2 - 4 5 )

G 9 9 9 5 E 2 - E 2 - 2 - - E J 2 2 - 5 ? )

G ) ) ) J 9 ) 2 2 2 2 ? 2 E - 2 J ) E - 2 2

) 4 - ) 2 - 4 5 ? ) ) 4 9 G ) - 2 - 4 5 ? )

E ) 2 2 - ) 2 2 G 5

E E E ) 5 2 - ) 2 - - E 2 E 9 E ? 2 E

G ) 4 J E E 4 J E ) 2 2 - ) 2

E E E ) G ) 2 J ) 2 2 2 J 2 ) ) ) E - 4 ) 4 G

2 - ) 2 J 2 4 G 5 - 5 J ) - )

? 2 E 4 4 G ) 2 - 4 J 4 E 2 - ) 5 ? E E - 2 ? 5 - 5 5 J 5 E 2 - 5 ? 9 E - G 5

9 ) 2 ) - ) 5 ) E 2 - ) ) ) - 5 -

5 E ) )

G E 9 E ) 5 2 - )

)

5 E - E E 9 ) 5 9 4 2 2 4 G 5 E 2 - 2 - )

Page 154: Goldrei.classic Set Theory for Guided Independent Study

� " # $ & " " & " + $ / 0 $ " # $ # & $ & $ & 0 + $ � 0 " # $ & $

" 0 ; $ " # " # + / $ " # 0 / ; 0 0 " $

� $ $ / 0 + $ $ & $ + ; " $ & � + $ " $ $ " # / "

# + + 0 " / ; $

# $ $ # $ & $ 0 $ " # $ + $ + / " / ; � 0 0 " $ / " $ " $ &

� + 0 � $ / 0 $ $ � 0 / / � $ " / & � 0 " / 0 $ + 0 � $

" # $ $ � + $ & # � + $ # / " # $ 0

/

+ # /

# � ; $ ;

& $ $ $ & " # /

/ +

/ 0 & # � # $ / 0 0 ; $ " # / "

+ " # " � + 0 / ; $

� 0 / ; ; � # / 0 / & $ � $ + / / 0 &

" # $ & $ + ; " ; ; + & " $ / " " # / # $ & $ � " # $

& " / 0 � " # � 0

� " � $ $ " + $ " $ / � ; / & � " & $ " $ 0 $ / � $ " � 0 &

" ; +

" + $ + " # $ " # / " � � 0 $ $ $ " � 0

# $ � $ " 0 � 0 " # $ ; / + " + " � 0 / $ � $ # $ ; ; � 0 0 " & " � 0 �

$ " � 0 + + # 0 " # / " # $ + $ + $ 0 " / ; $ � $ + $ " $ $ " / ;

" $ & $ � # $ " � / 0 $ $ & # $ ; ;

� 0 # � + $ � 0 / ; ; & � 0 � " $ + $ + # / $ $ $ 0 / ; $ + # � + " # $

/ + $ / ; 0 0 $ + $ + # $ / 0 $ & � / & $ + 0 � 0 / + $ 0 $

Page 155: Goldrei.classic Set Theory for Guided Independent Study

� � " # % & ' ( ' + # % # 0 ' 0 ' # 0 ' " " 0 % & " 0 % ' ( ' " + ' 0

% & ' 0 ' % ( ' � % + '

+ # 0 ' %

& # + % % & ' ' ' # ' 0 % " 0 ( + ' # % ( " # 0 + # ' % % " #

" % & ' & + ' 0 ' ' # ' " # # " % ' 0 ' % 0

0 % ( " % � " # ' + 0 " # 0 " ' % ' ( ' + ( ' � % 0 " ' ' # % " ' 0 " # # " % ' 0 ' % 0

( % & ' ( ' ' ( ' " 0 0 ' " + � 0 " ' + 0 " 0 " # + ' % '

'

' # ' ' % 0 % + #

% " 0 ' 0 % 0 % & + % % & ' # " #

" 0 # ' # ' & +

0 0 ' % & % ' & ' " + " % " # #

+ 0 ' %

0 " 0 # % % & ' ( ' % ' 0 ' " 0

0 % & ' # ' # % " ' ' % 0 ' ' +

# % ( " % # ' 0 % & + ' % + 0 % & " 0 ' + # 0 % & + %

0 + + " # ' & ' + # % ( " % " " 0 # % # % # % + " ' % " #

# % ( " % " # ( 0 0 % " # + 0 ( ' " ( '

# " # # " % ' 0 ' " & " 0 # # % + � ' " 0 + " ' (

# % & ' # ' % 0 ' % " ' 0 & + � � " # ' 0 " % ' 0 ' # # % + � ' 0 ' % 0 + # 0 % + ( %

� + % + # ( " % & ' % ( # + � 0 % % ' # % 0 ' % " # � ' 0 � + %

# + � # # ( 0 " # � # # + � ' ' % " ( 0 + � � " # ' ( " 0 ' '

& ' % & + % & ' # " # # " % ' � + # # % + � ' 0 ' % 0 " 0 # + � ' ' ( ' " +

' � ' % ' # " # " 0 ( ' 0 % + # % 0 (

' % ' + # % � ' 0 ' % ' + ' � ' ' # % & " 0 " 0 ' � # � ' ' #

" 0 # % + � ' " ' # % + � ' # " # # % + � ' 0 ' % 0 " 0 # % + � '

Page 156: Goldrei.classic Set Theory for Guided Independent Study

! " % & ' ( % ' + ! ' % & % " ' + ! % ( % % ! 3 4 4 + ' " % % & " 3 & % ! ' 3 4

+ % " + ' 3 ! ' % ' % % + " 3 % ' + 3 (

% % & " ' ( % + " ' 3 4 + ! ! + % " ' ( % & %

+ 3 % ' ( % ' ( % ! + ! % ! %

" 3 + % ' + ! ' ( 3 ' % & % + " ! ' 3 4 %

( ' 3 ! ' ( + ! " 3 ' ' ( + & ' + " ( 3 ' ' ( % & ! % % " 3 + 4

& ( % + ( % 3 ! + ' ( + " % ' + ! % ( % 3 + %

3 ( % & % % " % % & + ' ( % &

+ ( ' % ! % % ' " % ' & % ' 3 ' 3 ! ' 3 4 % ! + ! 4 % % ! '

" % " " ! ' 4 %

3 ( % & 4 % " % ! % " 3 3 ' % 3 ' % & % + " 3 + % ' + !

% ! % & 3 4 ' ( % & % + 4 4 % % % & + % + % ' % % ! 3 !

' ( % ! 3 4 % % " ( 3 % ! % ' ( " % ! % ' % % & % 3 (

% % ' " ( 3 % " + % ' " & ' & % ' ( % ! ( % & % %

% " % + ! % 3 ( ' + ! ' ( % ! 3 & " ! % % %

( " % " 3 % " ' 4 % 3 % ! ! ' % % % ! + ! ( ( %

! + ! ' 3 4 3 4 % ' " + " 4 4 + ! % ' + !

4 " ( ' " ' + 4 % % % % % 3 ( % 4 % % ! ' % ' ' % & % %

" 4 % % ' + ! " % ' % % ! ' ( % 3 ' & 3 4 ! % & 3 ! ' % " % ' 4 % "

% 3 ( % % % " % ! + % " & & % % + 4 4 3 3 + ! % 3 %

' 3 4 3 ! + % " % 4 4 ' ( % % + % ' + "

! 4 & % & ! ' ( % % 4 3 ' ! " ( + % ' % % ! 3 ! + ! + % " % !

% & % ! ( 3 ' + ( % ! + " + ! + % & % 4 % ! % & % +

' & % + " ! ! + ' % ' ( % ! 4 & ! ' 3 ' ( % 3 % ' +

% " 4 ' % % 3 4 & ! + ! ' ( 3 ' + " + ! ! + ' % " ' % " ! ' + %

+ % % ' & & % ' % ! % 3 ! % ! ' + ! &

% % ! ! ! + ' % " % ' " " + ! & % ' ( 3 ' + ! "

' ( % & % + 3 + % ! ' 4 + ' + " ' % ! %

& % & 3 ! ! % ' 3 4 ! ' ( % 4 + ! % ' ( % &

( % & % 4 4 ! % % ' ( % 3 ' ' ( 3 + + ! ! ' % + % ! ' ! + %

% % & %

Page 157: Goldrei.classic Set Theory for Guided Independent Study

; < > ? ? > ? ? > ; B B ; R ; S < S ; B X Y ; B ; ; Y S B Y >

X ; R < < ; X S B

> ; B ; > Y Y Y S B ; ; B X > < < S Y B B S < ; ; S Y S Y Y S B ;

S Y < ; B � ; R ; < ? B X ; R S < ;

� X S B � X B S > B ; X Y ; ; X B ; > X R ; < B X ? < S ;

S S < ; ; S Y Y S B ; B ; Y S < S Y Y B ;

X R ; > � X � ; ; ; R S < ; ; ? X S B S < X < S ? ; B X > X S ; X

X R B ; X < ; X B < ; R S ; R Y S B ; ; Y X Y B R ? Y

> Y X R ; R ; B ? ; X Y B ? Y ; R X < X ; < ;

> ; R ; > R ; > � ; ; B X > ? ? Y S B ; < ; ; Y ; < X Y > R > ?

Y ; R � X � B B � ; B S X Y > ? ;

> � X B � > B � ; ; > ? ? ; ? ; ; Y B < < ; X S < X Y > ?

X B > B X B ; ? ? ? ; Y X S

X Y > ? ;

X B > B B ; < B ? S B < X X Y B ? ;

< ; ; X ? ? S Y ; B X B ? ; ? > Y X R Y

X ? ; Y ; R S S B X ; S S B S < > R X X B X > X ? Y X S > ? ; S X Y

; R ; ; > � X ; S ; Y < R > B S X Y > ?

> � X B � > B X R > ; B ; ; B X > ? ? X X B X ? ? X ? Y S > ? < X

; R ; ; S B � > B S X Y > ? X ; ; Y B < S Y B > ? ;

� X B � B B � ; < ; B X > ? ? > ? ; > S Y ; R < S X Y B > ? ;

; Y ; > S ; X Y R X X B ; B X Y Y ; B S R X Y ? S B

? X S B B ; ? S < B X R B S X Y > ? Y ; R S ; ; X X Y < B R Y X Y B X B S Y

X X

; > B > ; ; > S > ? ; ? < X R ; < S Y X R X R > ? ? Y ; X ; X R

< S Y � > B ; > X < ; S Y B ; X Y ? ; < ; B

X R < > ; S ? B R X B ; < ; > S > ? < R ; B < > Y B � ; X ? <

> Y

� X B > B B � ; ; ; X Y B > ? > Y X ? X X Y ; R ; S > ? S Y

; X > ? ? Y > ; S < X B ; X X ; > B S X Y ; B X

; < S ; > B X < B X Y B > ? > Y < < ; B X S Y Y S B ; < B Y B X

B ; X R ; < > < B � > B > Y > Y X Y B > ? > Y < ; B S Y > S ? S B B X

; < S ; > ? ? X B ; < ; B < X S Y > S ; > S X X < ; X R B B

? S > B S X Y X X B � < ; B B ; X R Y X

Page 158: Goldrei.classic Set Theory for Guided Independent Study

5 6 8 : < > 8 6 @ C > C 6 5 I 5 : > > > S 6

8 S 8 : 8 6 C I 6 6 5 I > I <

8 : > S S 5 8 5 > : I 8 8 5 > > I : 5 >

C I > > I 8 I > : 6 5 < 6 6 : S > 5 I 5 I 8

8 < : 6 6 : 8 8 5 < : 5 I < < > 8 > : 5 S S 8 : > > : > : >

@ 8 I 8 : @ 5 @ 8 < I 8 S @ 5 8 8 < > > 5 I : > @ 5 : 8 I 8 :

C I 8

C 8 I 8 5 : 8 > 6 > 5

8 I 8 @ > 6 5 6 6 5 5 > > 6 5 6 6

8 8 S 5 > > > >

> C 5 I 8 < 5 5 5 > >

I 8 : I 5 : > 6 > 6

5 I 5 : > : : S I 5 I

5 5 I 5 5 I C C > C C 6 5 I I

> > : < 8 > I : 8 C 5 :

5 @ 8 < : I > 5 : : I 5 C 5 : 6 >

@ > 6 : @ 8 I 5 @ : I C 8 S C 8 > I 8 : I @ : 8 > > > >

I 8 8 : C I : 8 > 6

5 I 8 < < > S 8 > 8 6 8 > 8 S C I 5 : 5 6 6 C C 5 > : 6

> : @ C S S 5 C 6 S 5 I : 8 I 5 I > > I : C 6 5 I 5 : 6

C S 8 8 > C 5 > C 6 I 5 6 6 5

< 8 C S : S > > <

5 8 > S 5 5 >

> @ 5 > 5

C I : C 6 :

5 5 > 5 : : 5 : C I I 6

> > C : 6 6 I 6

@ 5 8 : > 6 6 5 : S 5 5 > :

I 5 6 : S I > : > I 6 > > 6 6 6 S :

S 8 < 6 : < > 8 5 > I S 8 6 6 S 8 6 : > 5 C 6 5

: @ 8 S 8 < 6 : C S 8 C > : < C 6 6 > 5 : > > 8 I 5 I

S 8 C 6 I 8 < 5 C S > : 5 < : 5 > I 6 6 S > > 5 >

S < > 6 6 6 8 : 8 6 C 8 8 < 5 < > > 5 > I 5 S I 6 > >

< > S : < > 8

> S C I 8

Page 159: Goldrei.classic Set Theory for Guided Independent Study

# # & ) + , - )

A # , C + D # - #

,

,

+ , C # & - D - + & + , C & , # + , - )

) + - C + # - A # D - A # & ) # # � # ) + � C + & # C

- ) C # +

A - # , A # + � D & + & , ) & + # # - ) # + # #

& ) ) # A , ) & - + A , -

# + C ) - , # C , & # + - A # & � ) # A + , ) �

) - # A , # + & + , , ) , + , ) + & � ) , + #

& ) + - A C - # ) + + # - � � - , ) + # #

) � #

- C # & - ) + � ) � + A � C ) # , # A �

C

+ D # & # � , ) & # A , ) & C # - ) # # C & � �

A # A # C # ) + # & D C D A # & � + # , & � ) C , - )

& - + # C # - # A A ) - + # # # A # C # ) +

A + D # A + ) # # ) , A # # # & � , ) , + # - + ,

) &

Page 160: Goldrei.classic Set Theory for Guided Independent Study

# & ( & ) * & , ( & *

|

� & ( & , # ( � # & & �   & & ( # (   * � & # | & & | �

& ) � ) � & � & | ) , , ) &   # | ( ( # � & ) , # * ) ( �   # &  

# � ) & ) * | ( � & , |   � # ( � * | � | ( � ) � � & , |   � # ( � ) | (   ) , , * & ) , | (

�     &   � & ( � ) & # � � ) � )  

& � ) & � � ) � | ( # ( ) , &

| ( � & # , # | (   � &   # � � & # # # & # * &

) # # ( # � � ) � � & * & ) & | ( ( | � & , ) ( & ) ,   | ( � � & | ( � & * ) ,

    | � ) � & * & ) & # ( � ) , | ( | � & , & ,   | ( � & | ( � &

) ( , |   � &  

| � � & ) & & ) * | ( # * & , # ( &

) # ( ) | (   � � & * ) � | # ( ) ,   # * & ) � | � | ( &

& � #   # � # � ) |   | ( | � & & |   &

  | � � ) | ( , | ( ( | � & � � & & & # (

� & ( & * & , | (   # * & ) , # ( &

� � | ( � � & ) & � & , | & * # & � # # � & *

| � & # � � & � &     # & � � � & ) * | & � #

( # � ) � # ( &   � |   & )   | & * � # * & )

� |   ) � � * & ) * & ( & , ,   ) | ( # ( � ) , & # &   ( * & , ) &

� � & | & # # � � # ) ( # � � & * & & ( # � & ( & � � & # * & � # & & *

# & � �  

&   ) , ,   & � � & � & * &   � & | ( � � & # & � & ) � ( � � & * & ) ,   )   | & (

& & | (   �   &   � &   #   � # � � ) �   & ( � | #

& ( &

|   , , # ( # ( &

& & # & # * & # � #

# , & � & � � & # & ( & & � #   # � ) �

Page 161: Goldrei.classic Set Theory for Guided Independent Study

" # " # " + " - / 1 / # + 4 5

/ 1 5 / "

/ # � " - � + - 5 + " " # " # " # " + # - / #

/ + # + 4 5 / � " - � + 5 / # " / # " 5 / - � + -

+ � / � - # - � " + / " 1 / / + " " # -

� / - � + - # - � " + / " 1 / / 5 / # " / # "

/ - � + - " � + " / # # " - " - � " 5 " 5 / - / 1 / 1 + # 5 " - 5 � # + -

" + - 5 + # 4 " - # " 5 - + - " � / - � " 5 " 5 / - � " 1 � 5 " 5 / + 1 "

" 4 1 " + / 1 # 5 - + # " - + # � " + 1 " 4 4 "

" 1 / 5 - � � 4 " 5 " # " / 5 - � / / 5 +

- � " 5 " - / 1 / # + 4 # " 1 / � " 5 " - / / 4 " # " 1 5 / 1 - "

5 " - + 4 4 1 " + 4 " # - # 5 / + " + 4 + + 4 " / 1 - � " # " - " " 1 5 " - �

+ / / " / - " 5 5 4 " / # 5 / # " 5 " 5 / " 5 " 5 " - 5

+ / 4 " / # + 4 " # / - # / -

� # / / # + 4 "

5 5 5 4 " - # / - # / -

+ + # # / - " / # + 4 " - " 1 " / # - + 4 " - " # + 5 " + 4 5

/ 4 " - � " # / # / / / # - + 4 " 5 " - 5 � " # " / # - + 4 "

� 5 / # - - - � - # / # - + 4 "

5 # / # - + 4 " + # � # / + 4 "

# / 1 � " / 1 " - 5 - � - 5

/

" - - 1 - / - " / # - � " " 5 / - � " 5 " " 5 5 / " / 1 " 1 5 - / # " 1

" � + " � / # - � + - + # + " � + " + #

" 4 / + " - � " " / 5 / " � " 1 " " - " " # � / " / + # +

" " / / - � - " " # - 5 # " " 5 - + / # " - "

/ 1 / + / # / # " # - / # 1 / - /

# + / # / # " # - / # 1 / /

/ " + # + 4 " 5 / # 4 " - � + # # - " " 1 5 " 5

5 # / - # - 4 " - � " " 5 / # - " # 1 " 5 " - " 1 + " + "

5 - 1 - 4 " - " " # - � / 5 " / + # " " - / # / � " � " 1 / # - #

Page 162: Goldrei.classic Set Theory for Guided Independent Study

! " " ' ( ) ( ' . / . / ) ( / " ( 3 4 ) / ' " ( / ' ( 3 " 4 3 ! ( 3 )

' . ( 4 ' ! 4 ) 4 " ( ! 3 / ) / " 3 4 " ' 4 3 4 3 /

) ' . ! ' . ( 3 ( ) . ) ) ( ' ! " / ' ) 3 ( ) 4 q / 4 " ! ) 3 4 4 ( q ( 3 ' !

4 ( 4 ' . ( 4 ) ' / 4 3 ! " ' ! ) ) 4 3 ! . ( ! / / ! " ) / " ' . ( . ( " ' 3 4

. ' ( ) ) 3 4 ' . / ) / ) ) / 4 " ' . ! 4 ( ) " 4 / ) 3 4 ( ' ( 4 " ' / "

. ' . ( ) ) 4 ) / ) " 4 4 " ' ! q ( / ' ) ' ( . ( ) ( ' . / ' / )

( / " ( 4 ) / . / 4 / q q " 4 ) 4

) ! 4 " ( ( " ( 4 . ( 3 ( 4 ) ( ( 3 / ) ( ( " 4 ' . ! ' 4 " ! / " ' 4

) 4 " 4 " ! q / " " / ' ( ) ( ) ! " ( ( q ' '

/ q q 4 q 4 3 4 ' ( " ( ' ( 4 3 (

( ( ! ( ' / ' . " ! / " " ' ( ) ( ) ! " !

. ( "

3 4 ( ' . / ' ( 4 3 ( " ' ) / ) q ( ! 4 . 4 ' 3

/ " ' . ( ( " ' / ' " 4 " 4 " 3 ( / q q 4

4 ' . ( ) ( ' 4 / 3 3 ! ' / 4 " ! q / ) " 4 ' " 4 " ' q / / ( / " ( 3 4 / '

) ' . ( 3 ( / ' ( ! q ( " ( 4 3 / " ' 4 " ! ( 4 . / / 3 4 (

4 " / " . ( ) / ) ( ) ! q q / ( 4 ) 4 ( ( ( 3 / ) ( / " ( ) / ' / " ' / ( )

4 ! 3 / 4 ) / " " / ' ( ) ) ( ) 4 . ( ! " ( ) ! 3 ( 4 ( ( 3 ' . ! ' ( / q q (

( / " ( 3 4 / ' 4 3 ( / ( / ( q ! ( ! q ( ) ) . ! ' 4 ! q q

4 ( " / " ' ( 3 . ( 3 (

! " ! / ( / 4 " ( " / ( 3 ! " / " ' ' ( 3 ! 3

/ ( 3 ( " ' / ! q ( " ' / 4 / q q 4 3

4 ( ! '

( ' q ( ' ) q 4 4 ! ' ( 4 / ) " 4 " ' q ( ) 4 3 / ) / '

( ) ( ' . ! ' ' / q ( 4 ! ( ( 4 " " ( ' / 4 ( ' ( ( " "

4 3 ( 3 ( / ) ( 4 . ( 3 ( / ) " / 4 ) / ( ' / 4 " ( ' ( ( "

! "

4 ' . ( 3 4 q ( / ) ( ! q ( " ' ' 4 4 ! 3 / " " / ( ' ( 3 ( ! q q / " (

" ' ( q ( ! " q ! " ( 3 ( ( 3 ( ! " ( " 4 ( ' / 4 " ( ' ( ( " ' . (

' . ( 3 ( / ) ( ) . ! q q 4 ' / ) ' 4 ( ' 4 ) 4 4 / . ! ( ' . ( / 3

' 3 ! 4 "

Page 163: Goldrei.classic Set Theory for Guided Independent Study

# $ & ' $ # + - ' + 1 ' 2 & 2 + 1 ' + 6 + & 6 2 $ + 1 1 $ 1 2 + 1 ' 6 1 ' $

+ # $ # & $ 1 ' 2 $ - 1 # $ + & $

$ 1 $ 6 - + 1 6 + T + ' - $ 1 1 2 + & $ 6 '

6 1 - 1 $ 1 $ & $ ' ' + 2 # 2 # $ - & $ - 6 T $ 6 1 + 1 '

# $ & $ # $ ' 1 6 & $ - - 2 - 1 6 + 2 # $ 6 1 - + 1 ' 6 + 2 # $

' $ + & $ & & ' - 1 $ & & - 1 ' - 1 2 $ 6 # $ & $ 1 $ $ ' ' 6 & # $ 1 $ 1 $

2 + $ 2 $ & $ 6 T 1 $ & - 2 # $ - 6 T $ 6 1 ' - + 1

2 $ & T $ 6 - 1 # $ - 2 ' + 2 # $ $ 6 1 ' - + 1 ' + 6 1 2 ' $ 6 ' 2 + # $ 2 # 6 2 -

+ 1 $ 1 $ ' + 2 # 6 2 $ - ' $ $ # 6 $

' + $ 6 1 $ $ 6 2

- ' $ 2 & # + & 6 & 2 + ' # + 2 # 6 2 + & - 1 ' 6 1 $ ' - 1 2 # $

2 + 1 $ 1 $ # - # - T $ 6 T + 1 $ + 1 $ '

- 2 + T T + & + 2 # $ # & $ & $ ' $ - 1 # $ + & $ 2 # 6 2

# $ ' $ + 1 $ # 2 + 6 $ 1 1 - 1 ' $ + + $ + # $ 6 ' - 6 2 ' +

& + $ 6 & - $ & ' $ 2 - + 1 ' 6 + T + '

6

6 '

6 # $ 1

# $ + & $

# $ & $ 2 + ' $ 6 + $ - - $ - 2 + 1 + 1 $ 1 2 - + - $ ' + 1 $ 2 # $

- 6 2 + + # 6 - 1 + - ' - $ 2 # $ 2 & 2 & $ + 2 # $ ' $ ' - 1 + $ # $ ' $ + 1

$ 2 # + # 6 # $ 2 2 - + 1 + T + + - 1 T - $ 6 6 T $ & 6 # $ & $ + 1 $ $ T + 2 ' 6

1 $ + 6 $ 1 2 T $ ' T 2 ' 6 + 2 1 T $ 1 2 # $ 1 + +

2 + + 1 2 T $ $ 2 - ' + 1 2 6 T $ 6 T + 1 - 2 # ' + $ $ ' ' $ 1 2 - 6 T T T T 6 2 ' 6 + 2

- $ 2 + 1 ' # ' $ 2 + + $ ' + 2 ' $ $ + $ 2 2 - 1 1 # $ & $ 2 # $ 1 $ $ -

+ 6 1 6 6 - 2 2 + ' $ 2 # $ # $ & $ 1 ' 2 $ - 2 # $ + & $ - 1 2 $ + T T + 1

$ $ ' $

# + 2 # 6 2 2 + $ 2 # + + 2 # - - 2 # $ & $ ' ' - 1 2 # $

# $ $ 1 2 1 2 # $ + & $

+ 1 ' 2 & 2 + + 1 $ 1 2 - + 1 ' 6 1

' $ 6 T $ & 6 ' - 2 6 T $ 6 2 ' + 2 2 + ' # + 2 6 2 6 1

6 $ & $ ' $ & $ 1 $ & 2 # $ - & $ $ 6 1 ' - + 1 ' 6 + - 1 & $ & & -

' 2 # ' ' - 1 6 2 2 # $ + 1 - $ # 6 $ 1 2 + $ 1 $ 6 2 - $

Page 164: Goldrei.classic Set Theory for Guided Independent Study

" # $ # $ # " # 8 # :

F

F T : U U 8 # # # $ U U #

# F ^ _ : F $ U

F

T U " U F T U " # U : $

: 8 ^ U U " # # $ F U : $ : T # U U F $ U : # 8 # # $ U U #

$ T F $ ^ $ U " # $ T " F U F $ # # 8 U " F $ F $ # # $ F U F : $ ^ : # ^ _ : $

U T " : _ # _ # U " ^ U F T : $ # : $ #

: # $ # ^ : _ : T F # $ # $ # ^ : # T : $ F $

8 # ^ _ # F U " # F ^ _ # ^ $ T F : $ : $ T F T F $ : # ^ 8 ^ U #

T T : U " U # U " # F # $ U F U $ U F : $

^ T U :

: $ # ^ T

$ : ^ T : 8 T ^ T

: _ _ F T U # 8 # ^ _ $ # 8 " # 8 #

F : T : #

: U " # #

" # $ # $ #

: T " : _ " # " ^ T : $ $ #

F 8 T U $ : U # " " #

# $ # " # 8 # T _ U : # 8 F T # U : # # U " ^ U

"

Page 165: Goldrei.classic Set Theory for Guided Independent Study

� � " � $ % & ' � � & ) % & & $ % $ ) ' % ' � � " ' � � & " ) � $ $ & ' � &

% � ' 9 & � � �

) ' " & % ' � " & �

% " ' � ' ) $ ) ' � � & � % 9 ' " � & $ �

� $ % �

%

% " ' � '

" ) ) " % � " � " & " ' � � & " ' � � � � ' � " % ' " & � % " 9 � ' � � 9 $ % � $ % �

� & $ % � � � % � " ) ' ' ) " ' $ ) ' '

" % � ' ' � ) � & $ " ' � ) " & � & $ '

� " ' �

) " � � � � $ ' � ) $ ) & " ' " & $ 9 & $ ' � � ' $ � � � 9 " ' %

" & � % 9 % " & $ ) ' � � � " % � ' % ) � & � � & � 9 $ " ) % ) ' $ % �

' � ' $ ) ' & � & % � & � 9 ' % " ' & � � ' $ � 9 � � $ % " $ ) ' ) $ " ) "

% � ' ' � � $ & & ' � $ ) " ' ' � � ' " ) ' " ) & " " ) � ' " ' � � " ' �

% � $ " ' $ " ) 9 ' $ 9 $ " ) ) � " ) � ) $ ' $ " ) " ' � $ & % $ � '

" " & % � � � & ' � � & " $ ) ' � $ � " % � ' 9 9 $

$ ' � " ' % ) ' � � % � ) & � 9 9 " $ ' $ ) $ ' " ' ' �

& � & � % � ) ' $ ) ' � � % $ � " � ' ' % 9 $ 9 9 9 � ' "

� ) � & " � % $ 9 % " " % � � $ $ 9 9 � ) 9 � ' " � ) � )

& $ � � $ " ) � & $ ) 9 " 9 � ' % $ ' � 9 ) � & " � & $ ) � � 9 9 " �

" � $ ) 9 $ ) � & 9 $ � & 9 � % % � %

' � � )

) $ ' � ' $ % & $ � � ' $ � % � 9 9 � " � % " � " % " � " $ ' %

� & � ) � % & " ' � & $ ' � � ' $ " " ) � � $ ) " 9 � ) ) $ ' � % � % " & $ ) % ' ) �

) 9 ' $ ' $ " ) " ' $ ) ) � % � ' � � � " ) � % " & � % ) ' & " �

% � ' " $ � % $ � ' ) " ' � ' � " ) � % � ' & ' � $ � � )

� " � '

� " ' �

� % 9 ' 9 $ � ' � � % � � % ' ' � ' $ ) ' � � & $ ' � � ' $ " $ ) $ ' � & $ ) 9 % � % � " 9

� �

� � & � � ) % ' � � & � ' � % ' & � 9 ' $ � ' " ' � � " & � & ' $ ' " ' �

) ) " % " ' � ' � & � $ % & � ' � ' " ' " & $ ' & & � ' % � � " ) 9

Page 166: Goldrei.classic Set Theory for Guided Independent Study

" " $ " $ $ ( $ ) + , " + $ " " . + " " " 0 " + ( 2 2 4 . $ " 4 $ ) $ " $ $ 2 0 + , .

" 0 ) 4 4 , " ( 2 " + 0

0 4 . " 0 " 4 . + 4 " , " $ . ( ( 2 0 " $ + " . " ) . " , ) , 0 " , $ 4 ) 0 " ) + 0 4 ) ,

" ) 4 . 4 . " ) , ) ) , + 2 4 . 4 . " $ " + 0 " + 0 4 " , " + ( ( " $ , "

+ 0 4 ) , $ , 0 + , 2 4 . " 4 2

" " + ( ) 4 . " ) ( ) 2 0 " " , 2 $ " $ + $ ) 4 ) ( ) + " 4 . " $ 2 " $ ) $ " 4 $ ) ) , " ) ,

( " $ $ + . " + 2 + ( $ 2 0 + " ) " , " 2 , " + 2 " " ) + 4 " + 4 2 + ( 0 )

" " ) ) 4 $ 2 " 4 . " ) 0 0 " $ ) $ " . " ) , ) + 0 ) 2 ( ( " " 4 ) $ "

4 . " , ) " , " , 0 $ 4 " 2 0 " ) , " 4 ) , ) " ) , , " $ ( 4

) " 4 + 2 $ + $ " ) , . .

+ 4 " 2 0 0 2 "

) $ " . + 4 $ 0 4 " ) 4 . + 4

2 2

. + + , " 4 . " $ 2 " $ ) 4 " ) ( ( ) 2 0 " 4 $ . + 2 $ + , " . " " 2 0 " , ) $

2 4 ) $ ) " ) 4 . " , " ( ( 0 ) 0 $ "

+ . " " 4 ) + ( ( 2 0 2 4 " $ " 4 $ )

. " $ " 4 ) + ( ( 0 2 4 " $ " 4 $ )

. " $ " 4 ) + ( ) 0 + ( 2 0 0 " $ " 4 $ )

. + 4 + , " 4 . " $ 2 " $ ) 4 . " ) ( ( ) 2 0 $ " 4 . + 4 2 $ + , " 4 . " " 2 0 " , ) $

2 4 ) , $ ) " ) 4 . " , " ( ( 0 ) 0 $ " 4

+ . " $ " 4 ) + ( ) 0 4 0 ) 0 4 ) 0 $

. " $ " 4 ) + ( ) " 0 $ $ " 4 $ ) 2 4 . 4 . " $ + ( 4 ) ) ( )

" $ " 4 ) + ( ) 0 " " 0 $ " , 2 " $ ) , " + ( 0 " , $

0 2 ) ) , $ $ 4 ) " . " 0 ) + ( (

) , , " $ " 4 2 " (

2 4 . 0

2 2 + , ( 2 $ 2 4 ) " . " 0 ) , + ( (

2 4 . " 0

. + 4 + , " 4 . " $ 2 " $ ) . " ) ( ( ) 2 0 $ " 4 $ . + 4 $ + " 4 . " " 2 0 " ) $

2 4 ) ) " ) 4 . " , " ( ( ) 0 " 4

" $ " 4 ) + ( ( 2 0 , " + $ 0 0 2 ) 0 $

Page 167: Goldrei.classic Set Theory for Guided Independent Study

% & ' & ) + , , & / & + ' 0 1 1 5 0 ) 1 '

% & ' & 5 ) + , , 1 / & + ' 0 1 1 ) 1 '

% & ' & 5 ) , & / & ' 0 1 1 0 ) 1

+ % ) 5 % + + 5 ) ) 0 1 5 0 1 5 % & ' ' & ) 5 % & , + 1 & 1 & ) 0 1 &

+ 0 / + / + / % 0 % , 0 & ' & 1 5 / & , 0 % 0 ' ' ' & 5

) 1 ' % + / + / & 5 % & / & % + 5 0 ' 5 % & ' & 5 ) + , , ' % + / ' 0 1 0 1

5 % & 5 ) ) 0 1 5 ' & 0 1 & / ) ' 0 5 % & , , 1 ) 1 ' & 5 , 0 & ) /

1 ) / / ) ) ) % & ) / & & & , ) 0 5 & + 1 5 0 ) 1

& 1 & + ' ) , , ) ' 0 & 1 + 1 ) / & + , ' + 1 0 1

& / & ) 5 % % & 0 & 0 + , & + 1 0 ) 1

% & / & % & ' 1 ' / & 0 0 5 ' 0 1 + 1 ) & + 1 ' 0 ) 1 ' + ) 5 % &

& ) / & / 0 1 ' ' 0 1 / & / / 0 1 ' 0 1 & + & / & 1 & & ' ' + / % & 1 & 1 &

5 ) & 5 % & / & + , 1 & / 0 % & 0 , & + 1 0 ) 1 ) + 0 1 & 0 1 5 & / , & + 0 1

% & 0 0 5 ) 5 & & + 1 ' 0 ) 1 ' ) + 1

, + % 0 1 ) + & 0 ) 1 0 1 5 % & / & 0 ' + / ) , & 0 % / & / / 0 1 '

+ % ) 5 % 5 ) / + , , 0 1

+ 1 ) / & ' , 5 5 % 5 + 1 + / & & ' & ' 0 & , ) & ) 1 & / ) ' 0 1

+ & 1 & ) / & , + 1 0 1 % + 1 + / & 0 & & 1 5 ) & & / 1 ) 0 & 5 0 ) 1

& & & 1 5 % & & ' 0 ' ) 1 5 0 1 ) ' % 5 % & ' & 5 ' + / & 1 ) 5 % ) & ) / % 0 ' )

5 % + 5 0 % 1 0 1 ' & 5 , ) ) 1 & + 1 & + , % 5 % + 5 5 % & + / & 0 & / & 1

+ 1 5 ) ' ) / + + ) 1 + 5 0 ) ' ) 1 & ) / ) 0 1 ' & ) ) , ) ) % + &

) & ' ) & 5 ) ) , ) + 1 0 1 + / 5 0 , + / % + & & 5 % & ) 1 & 5 ' ) ) 5 0 1 0 5

+ ) 1 1 & 5 & 1 & ) % ) , & + , & 5 ) ) 5 % & ) , , ) 0 1 & & / 0 &

% ) 5 % 5 % & / & 0 ' 1 ) ) 1 0 1 ) 0 & 5 0 ) 1 % & / & + 1

% + & % & 0 / + , 5 ) ) , ) 0 & '

% & + ) ' & 5 0 & 1 & + ' 5 % & ' & 5 ) + , , / & 1 & / ' ) 5 % & ) /

% & / & & + % + & ' ) 1 & ) / ) 5 % & / ) 5 % & + , & ' ) /

+ ) % 5 0 ' 1 ) 1 5 + , &

' + ' & 5 5 % 1 5 % & ) & 5 ) 5 % & 5 % & ) / ) & & ' & & ' & ) /

& , & , + & 5 ) & + ' / & & & & 5 0 ) 1 ) / + / 0 & & ' / 0 5 ) 1 )

5 % & & & ' & & + / & ) & 5 & 5 % & & 5 ) + , & & & & / + , &

' & 5 ' ) + 5 5 & ' & )

Page 168: Goldrei.classic Set Theory for Guided Independent Study

. / 0 2 4 2 4 : ; > 2 @ @ ; / 0 H . 0 ; / @ / 0 . : > > . ; > 2 > / 0

> H @ @ H ; 4 : . 0 > 2 ; . @ / . : @ / 4 0 2 0 ; @ / H ; > > ; 4 > . 0 > 0 4

0 @ 0 . > @ 0 H 2 @ H 0 H ; > . 2 : ; > 2 . 2 > 0 @ 4 > 2 4 0 0 . 0 ; 4 @ / 0 : >

4 @ 0 H . 0 . : : 0 H > 2 > : . / > : ; 4 0 > ; / 4 > > / > ; 0 H @ 0

> > 0 0 @ / 4 > > > 4 / @ > . : / > 2 4 . 2 > > / > 4 @ : : @ 4

0 H > 2 4 0 0 @ / > 2 4 ; / 0 H > : ; 4 @ . : 0 > 2 > . : H ; H : ; > ; / 0 H > @ > /

/ > 2 . : > 2 > > / 0 0 / > 4 ; 0 ; 4 > . 4 0 @ @ / 0 2 0 .

> . : > 2 / @ 0 ; / 0 > : ; ; / @ / 0 2 ; 0 ; @ / > 2 ; > . : : 0 @

> 2 4 . / ; / @ @ @ @ > 4 0 @ / > 2 ; / 0 H >

; 4 0 / H > @ > / ; / 0 > 2 . : : ; / 0 > / . ; ; /

0 H > 2 > . > / 0 0 @ / > 2 ; 4 > 0 @ @ 0 ; / . @ / 0 2 0 @ / 2

@ ; / 0 H ; 4 @ 0 . 0 . / ; 0 H . > 0 > 2 4 0 0 @ / > 2 ; /

H > : ; 4 0 ; / 0 H > @ > / ; / 0 > 2 H > >

. 4 4 ; / 0 H 0 @ / > @ > / 0 2 / @ 0 @ / > 2 ; / 0 H > : ; 4 0 . 4 @ > / ; 0 > 4 0 . > @

0 H > @ > ; / H ; H . 4 > > . / @ / 0 0 . 2 > . : / > 2 / @ 0 ; / 0 > @ 2 ; ; / . :

; 4 0

. H @ 0 H . 0 ; . 0 H > 4 . @ : @ @ ; / @ 2 0 2 0 0 @ / 4 ; / 0 : ; 0 ;

H > @ > / / > 2 . : @ / > . ; : @ / 0 @ / > H > / 0 H >

; 4 . 2 > . : / > ; / 0 H ; 4 ; / 0 > / @ 0 / : ; 0 H / @ / ; ; 0 . 0

0 H > : ; 0 @ 0 . ; / > . : : 0 > 2 > : / > 2 4

@ 4 > / @ 0 H . 0 . > > > / 4 > 4 : : @ / 4 0 > @ . :

/ 0 H > . @ > / / > 2 > 4 > > / > ; 4 4 0 2 ; : / 2 > . 4 ; / . / @ / >

. @ > . / @ 0 H 4 4 @ H . . : ; ; 0 ; > > 0 H > 4 > > / > ;

0 2 ; 0 : > 2 > . 4 ; / . / @ > > : @ . / @ 0 H > 4 4 @ 4 . : ; ;

0 4 4 0 . ; H @ 2 . 2 > . : . / . : ; 4 @ @ 0 H 0

; H @ 0 . 0 > . H

; ; @ H 0 @ 2 / 0 @ / 2 . 0 ; / 4 ; @ /

0 H . > : ; 4 > . : : > . : / > 2 4

@ : > / > ; . ; 0 @ ; ; 0 / @ 0 : > ; /

/ > ; > @ 0 . 0 > > 0 @ 0 2 / / > / . / > 4 > ; / > 2

@ 4 ; 0 ; 0 H . / : @ > > 4 > / 0 ; : : 2 > : ; . 0 > . / 0 @ 4

> / 0

@ ; : > H . 4 H @ ; 0 . 0 0 2 . / > / > / 0 : / > 2 4 > ; 4 4 > > 0 > > 0

> > 2 ; 4 > 4 ; / . ; 0 @ : : 4 H > / > . 4 4 @ / 0 @ > 0 . / 4 > > /

0 . : > 2 0 > ; / 4 ; / ; 0 > . : @ 0 @ 2 > . : : 4 > 0 2 . / > / > / >

@ H . 0 @ ; @ / > . / / ; / @ . / ; ; / > 0 > ; 2 . >

. / 0 @ 2 4 > / 0 ; / ; @ ; / 0 2 0 > 4 4 . ; . 0

. : @ 4 0 . : : > . : / > 4 2 > 0 2 . / 4 > / > / : ; 0 @ @ / 4 0 2 0 ; / . 4 ; / : > @ / >

@ 0 >

Page 169: Goldrei.classic Set Theory for Guided Independent Study

+ , 0 , 0 3 + 4 5 , , 4 5 , 4 5 3 0 4 8 8 5 3 9 : 3 , 4 0 4 8 8 , 5 , 0 @ 5 3 9 : 3 4 0 @ 3 : 3 3 ,

@ , 9 , @ , 4 5 3 9 : @ 5 , @ , @ 3 : 8

0 5 + 3 4 @ 9 9 : 3 4 0 @ : 8 : 8 , 9 : @ , 9 4 + : 0 8 @ 0 4 : 8 4 3 +

4 @ 3 , , 9 0 , 4 , @ 3 0 4 @ 4 : 8 , 9 , , 5 + 3 + : : @ ,

0 : @ , 9 0 3 + , 4 @ 4 0 8 @ 0 4 : 8 0 0 , 9 8 3 4 8 @

+ , , : 3 0 0 @ 8 4 8 , 0 , , @ 0 @ : 3 0 5 0 4 3 5 0 , 4 , @ 3 5

: , 3 , @ 9 : 3 4 0 @ : 8 @ , 9 , 9 ,

4 0 : 3

4 4 0 5 , 3 : 3 5 , : 8 0 9 3 , 0 9 , 0 5 0 3 :

+ , 9 , 4 5 3 + , 3 , 4 : 4 , 0

4 4 0 3 + : 3 : @ , , : 3 4 @ 3

9 , : 8 8 3 : 3 4 : 0 8 @ 0 8 0 4 @ 4 8 , 9 , , 5 3 :

4 , , 3 + : 3 4 3 + , @

+ : 5 @ 0 0 4 0 @ 4 @

, 3 , , @ ,

8 : 4 5 @ 0 3 4 0 @ 8

4 4 3

4 4 5 , 3 + , 9 , , 4 @ 9 , 5 8 3 5 3 0 , 3 : 3 4 5 3 9 : @ 5 , @ , @ 3 : 8

Page 170: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � ! " # � � & � ( � " # ! # ! � " + # � + # ! ! � ( + ! � (

! � � ( # " ! + # � + � � " � & & � + � � ! + � � !

! + ( ! � ( + � � ( # ! � & " # ! # ! � � + � ( � + � # & + ( � ( � " � +

� ! ( # ! � & # & & � � " � # � & " ( + ( � ( � " � + � � � ( � �

" � + " � " � � + + # � " � " � � & � � ! + ( # " � &

� � " ( � ( ! ! � ( � � � # & # � ( � ! + � ! + � (

� � � � � + # " " + � ( � ( " � ( � & " � & ( & � + �

( ! " � � # + ( � � � + � � ( � ( � � ( � ! " � � ( "

( ( + + � � + � + � ! + � ( � + � � � ( " ! + � " � � + " + � � " �

# ( � ( + # � � # # " # ! # " � � � � ! + � � ( � ( " ! ! # �

" � ( � ( + � � " # & � + � ( " + � � ( � ( � + � � ( + � ! " � + �

# � � + # ( " # � ! � � � � � " + � & " � + � � � � (

� � ( � � ( � & � ! � # # ! # " � # ! � + � � ! + � � � ( � ( " � ! ! �

( � + � # " � � + � # � " + # + � ( � + # � " ( � " � � " + � �

� & # ! + � ! �

� " � # # � & � � ! + ! � ( # + " ( � ( + � � + � � ( " � + " ! + � �

! �

� � � & � � + � ( � # " � ! � + # � " + � & � � ! + # ! + � � ( + # " #

� " ( � ( + ! + (

� + � � ! ! + # " ! � & � � ! + + � ( � # " � ! + � � & � � ! +

+ � # " # � & " + ( � ( + ! ( (

� � ( " � + # # " ! � � � " � & � � + � ( ! # !

� # # " + � ( � !

& & � � � � ( � ( + # � " � � � ( � " " # & � # ! + � ( " � � " ! � � " � +

! � � � � ( � ( + # � " + � � ( � ( ! � + # ! � # ! � � " & # !

� � " � � � ! � � + # & & + � � � " # � � ! " + ( + # !

� � ( # ! � " # ! � # " � � + � � � � & & � � ! # � ( � & � + + � ! !

� ( � " � & � ( � � " � " � � & & � & " # ! � " # � � � ! � ( # � � + #

( � ( � " � + " � " # ! � # � � # � + � & + # & + � ( � " � +

� � � ( ( ( � " � + + � " � & & ( � # ! ( " � & � " � �

� " # � # " � ! � � � � � � ( ( � ( � # � � # + � � ! � +

� + !

Page 171: Goldrei.classic Set Theory for Guided Independent Study

� � ! & ( ) � * � ( ! - & � ! ( & ( ! ( � ! � � ( ) ! � ( ( ) )

! � ! & D � � ! ( � � ! ( ( &

- D * ( ! & ) ! ( )

� D ( & ( ) � � �

- � - ( * * ( � ( ) ! � �

( * * � &

( ) D * * D & � & � D & -

( ) D * * D & � & � D & - &

D D D ) * ( ) ) - D � ! - ( * * ( � & � ! � ( & D * ) ( ) !

( ) D * * D & � & ( )

! � � D ( & D & �

! D * � ( � - ( ) �

D - � * ( D ( D ! - � ) * * * � & )

- � * ( D ( ! - � D ) D * * * � & D )

( ! - � � D ) ! D * ! & ( ! * D )

� & D & ( ) ) � ! D & � � ( � D � ( ( ) ) � & D & D ! ) D *

( ) * - D ! - ( * * ( � &

D ) ! � ( ) ( & & * ! D ! ( - & ! -

( ! ( ) � ! ! & D � ! - � ( �

D &

D � * ) � * * � & ) ( ) ( & D & * ! D � ! (

( ! - D ! � � D ! � D * ) � ! � * * � D ) ( ) ) ( &

( � & D ( ! D & - D * * ! ) D ! D � & ( ) ) � !

- ) ! � & ) � ( ( ! - ! - ( ) ) ( & � � ) � ! ) � ! � ( & ! ( ( - ( ) )

( & ( � & ! ! - � � � � ! - D ! � � � � D � D ( � ) & D ) � -

* � ( * � ( ) ) ! D * ! - � � ! � ( ( & ( - � D & D )

( ) ) � � � (

D * & D ) ( ) ( & D & ) ) � & ( ! - ! - � � ! (

( ) � ( & & ! ( D ! - � � ! ( ) ) � ( & � & ! ( - � - D ) * *

� ) & D � ! �

Page 172: Goldrei.classic Set Theory for Guided Independent Study

� � # � # # ( # ) * � , * - 1 * # � - �

# - * - ) (

) * # , ( ) * , � , *

* � , � � � # ) ( 1 # M � ( 1 # ( ) ( # , * , � , ) ( � � � M ) * , ( �

� ) ) M 1 � # ) # 1 # � # 1 * � * * M * , � # , ( - 1 � * , � ,

� M 1 ) * * , 1 � * - � � M # ) ) , # # - ( � 1 # � # ) � M # ) ( # ,

� * , � , � M 1 # ) ) M ( ) ) ( M M M # 1 * 1 ( � , � , # � , # - ,

( - # - � ( M # � # ) - � # , * � * - # - M * � ,

# ) ) � � M , � # 1 � � ) � � - ( - 1 � * , � , � M �

M M * # � ( 1 � � , � # * , � , * - ( M # * , , � M * - ( - * , �

1 � , � � # M ( * M * , M , ( * � # � # M * ) ) * M

� � # M � # - M � * ( M * -

* ( M ( # 1 � * ) ) * ( - , * � , ( �

* ) ( - M - * � � #

* , # ) ) # - ( - ( - �

M , ( # ) * , � , # � ) ) ( - # ( ( - # ) , � ,

* , # ) ) - ( - * , * ,

( M # - # - M

# * M � 1 # ( M � # # , # ) * , � , * - # - 1 # 1 � � ) # ( * - (

� - �

( # - * - ) ( # -

1 * 1 # ( # M , ( # ( # ) * , � , * - # - 1 # ( ( M ( - # ( * -

) - � # 1 � M * M

M � 1 # ( M , # , ( # ) * , � * - - 1 # 1 � , � ) # ( * ( M

� - �

# - * - ) ( * ,

* 1 ( M # � # # , # ) * , � , * # - 1 # ( ( M ( - # ( ( * -

) - � # � M * ( M

* � 1 � # * , � , * ( M # M * # � ( # M , ( * , � , * - # - ( �

� , M � # - M 1 1 � - � # ) # * # # , ( # ) ) * ) - � # , ) * � , �

M � � 1 ) ) � * � ( � , � � # * , ( * , * � * , � , � � -

1 M 1 * � * - � - ( � * , � ( M M M - * ) ) * ( - * , � , � ,

* - � - * - � M 1 # ) ) , � # # M , * , � , * - # M 1 * 1 ( ( ) � M M 1

, # � , 1 - � # � , #

Page 173: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � �

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Page 174: Goldrei.classic Set Theory for Guided Independent Study

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Page 175: Goldrei.classic Set Theory for Guided Independent Study

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Page 176: Goldrei.classic Set Theory for Guided Independent Study

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Page 177: Goldrei.classic Set Theory for Guided Independent Study

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Page 178: Goldrei.classic Set Theory for Guided Independent Study

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Page 179: Goldrei.classic Set Theory for Guided Independent Study

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Page 180: Goldrei.classic Set Theory for Guided Independent Study

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� � ! � � � � � � � ! � � � 7 � 7 � * � � � � ! 7 � 7 ! $ � *

Page 181: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � �

� � � � � � � � 3

3 � � � � � � � � 3 � � � � � � 3 � � � � � � � � � � � � � � �

� � � � � � � � � 3 3 � � � 3 � � � � � 3

� � � � � �

� � � � � � � � 3 � � � � � � � � �

� � � � � � � 3 � � � � � � � � � � 3 �

� � 3 � � � � � � � � � �

3 � � � � � � � 3 � � � � � � � � � 3 � � � �

� � � �

� � � � � � � � � � � 3 � � �

� � 3 � � � � � � � � � 3

� � � � � � �

3 � � � � � �

� � � � � � � � � � � � � � � � 3 3 � 3 � � � �

� � � � 3 � � � � 3 � 3

� � � � � 3 3 � � �

3 � 3 � � � � � � � � � � � �

3 � � � � � � � � � � � � � � � � � 3

� � � � � � � � 3 � � � � 3 � � � � � 3 � � � � �

� � � � � � � � � � � � � 3 � � 3 � � �

� � � � � � � � � � � � 3 � � � � �

� � � � �

� � � � � � � � � � � � � �

� � � � � � � � � � � � 3 � � � � � � �

� 3 � � � � 3 � � � � � � 3 � � 3 3 � � �

� � � � � � � � � � 3 � � � � � � � � 3 � � 3 � � � 3 � � � �

� � 3 � � � � 3 3 � � � � � � � � � � 3 � � � � �

� � � � � � � 3 � 3 � � � � � � � � �

Page 182: Goldrei.classic Set Theory for Guided Independent Study

0 4 5 6 8 6 ; 6 8 ; = > ? 4 ? 5 > 0 ; 0 4 5 4 6 8 4 = 6 > 4

5 > 0 4 ; > ? 4 ; 0 = 4 6 4 ; 0 6 8 > 0 >

; 6 8 5 ; 0 ? 4 ? 8 5 > > c 8 5 8 ; 0 > 0 = 8

8 5 ; 8 4 6 8 > 0 ; 0 > ? ; = = 8 > =

8 4 4 ; = 4 > 4 8 4 > ? = 6

> > 8 ? > 4 ? 8 4

; 4 0 5 4 > ? 4 8 6 8 ; 8 0 8 4 0

0 > 4 ? 8 6 > 0 4 = 8 8 6 > 8 0 8 0 c 8 4 4 > 6 ? 4 = 4 4 0 5 5 4 > ?

0 ? 4 c 8 0 0 8 6 4 = 0 = 8 8 6 > 8 6 5 4 > 8 ; 5 ; 4 > ?

4 6 ; 0 4 = 8 0 4 ; ? = > 4 4 5 4 6 4 4 6 4 > 4 > = = > 8 0

5 6 ; 6 4 4 0 6 5 8 5 ? 4 > ; = 4 ; 0 4 ; ? > > > 0 6 4 = ; > 8 6 4

; 8 > ? 4 ? 4 4 = 8 0 0 8 > ? 4 = 8 0

8 5 > ? 4 ? 4 4 ; = 8 0

5 > ? 4 ? 8 5 > > c 8 6 > 0 8 > = 8 0 8

> ? 4 ? 4 4 ; = 4 8 0

8 5 ? 4 4 4 = ; 0 8 > ? 4 ; = 8 6 0

; 0 > ? 4 ? 8 5 > > ? 8

4 6 4 ; = 0 4 ; ? = > ? 4 5 4 0 = 4 6 0 6 6 >

4 4 4 0 > 8 8 6 > 4 6 c > > > = = > 8 0

; 4 5 ; 6 8 5 4

8 5 ; 0 8 > =

8 5 6 4 ? > > 8 >

6 4 0

4 0 > ; 6 8 > 0 = 5 > 8 > = 6 ? > >

; > 6 6 4 > 4 4 6 ; = 4 > 4 0

= 8 ; 0 4 4 = 4 4 0 6 >

4 4 ; 0 > 0 4 6 5 5 4 6 > > 6 = 8 ; 0 > 0 = 8

> 6 c 6 c 4 > = = > 8 0 ; ? 4 ; 6 8 5 4

8 5 ; 0 4 ? > 0

? = = 8 6 4 0 8 0 > 0 4 ? > 0 > 6 ? 8 5 >

8 6

> 6 c ; 6 4 0 = 6 4 > 0 > ? 6 > 6 8 8 0

> = = > 4 6 > > 8 5 0 = 8 0 > ? 6 5

0 4 6 4 0 0 4 6 4 6 > ; 0 = 4 0 >

c > 6 ; 6

= 8 0 > 0 = 8 =

Page 183: Goldrei.classic Set Theory for Guided Independent Study

� � � � $ % & $ � * - % - � 7 $ 9 &

% 7 � 7 $ 9 % 7 $ 9 9 * - &

� � % 7 � 7 $ % & - \ 9 $ *

& % - � � * � � - & % � 7 % % *

9 * � * � % 7 - \

� * � 7 � & 7 � 7 $ 9 % * * � - �

% 7 \ &

- * % - � � � � $ & & � \ % 7 % % 9

$ $ * % � % \ \ % $ 9 � � � \ \ % 7 � 7 $ 9 % % 7 $ 9 9

� $ � 7

% � * � - & � % * % & - \ $ & & $ � � � % * \ $ & � - \ % 7

� 7 � � & � % � � � $ � - & % $ � * & � $ � � 7

$ & % 7 $ & $ � � � & � % * - & � % 7 � 7 $ % 7 * $

$ & - � � * � % % * � % � � - � \ % % & % � �

\ 9 $ 7 % % & � - $ 9 - \ \ � - & � � & �

% � � - * % � * � \ \ % &

- * � & % & � � 9 � $ \ & - \ % � 7 � - & * - 9 & � $ �

& % 7 & $ � � � � * � & $ & & �

* � & � 7 \ 7 � � � � % & � - $ & 9 - \ $ $ � % $ � * � & $ & & $ � �

\ $ & � - % 7 7 % & $ 9 � & 9 � - * - 9 � \ � % � \ � � & � � &

& & � �

� � � % & � \ $ & $ � 9 � � $ & � \ % � 7 �

- $ 9 % & � $ & � $ - $ � � $ % &

7 � \ - � $ % & %

$ - & & $ % - \ \

* * $ - \ & � - % 7 � % & & * - $ * � * $ 9 % � $ % 7

$ % * %

� 7 � % & & � - - 9 � � % % & 9 � 7 � � � %

- 9 � % * $ 7 � � � & � % 7 � % & % 7 � $ % % 7 * � %

\ & � - % 7 � � � * $ * % � * � % \ \ % $ & $ � * � � � $ &

� 7 %

Page 184: Goldrei.classic Set Theory for Guided Independent Study

# $ % & ' ( * ' * ( $ ( * . & / ( * ( * # # & ' ' * / % # ' & ( ( * &

$ . ( > * . & ' $ % / $ * * $ % # . # # * . * * ( # * $ % % & ' & ( *

$ ' $ . & > * ( & # # . * . * * / ( # & # . & % . ( ( & # # '

& . ( % * / % # # $ ( $ $ # $ & ( # ( ( # * . '

' $ ( . $ % * $ & / ( # ( $ * % % * % * ' # & ' & ' *

$ # ' $ / . * $ ( * . * % * * ' ( / % % ( * * % * * ' ( # & / & # .

# * # % * ' ( & * % * ' * * / # # * ( /

. * > * . * / / ( # ' * ' & $ % & & / ( # # & $

( # > * * % * * ' ( / & ' # # * # % ( & # ' ( # * #

> & > . * & / # & ' ( $ ' ( * ' * ' * * # # /

* ( $ ' * % & ' * . % # . * . * / * ( / * ' ' * ( > * #

$ ' . & ( * ' /

( # * ( > * % & ' * . % # . * . / (

& > # . * . * ' *

& '

& $ '

& $ '

$ > * > $ ( & ( > $ * ' * $ # * & / $ % & ' * % # . * * / *

# ( ( > * . * * # . * * * & ' # # ( $ ' & ' (

* ( / ' * / ( & $ ( * ( * ' / * ' & % . # . * . * ( $ * & ( &

$ # * % / # ( $ & ' ' * # . * * * ( * * $ ' / ( ' #

# * / # . & ( # * * ( * * # * ( * # . ' & * ( & &

# # ( & ' ' * # . * / & ' * * & ( & / ( * * ' ( $ % & ( * # & ( * * > &

# ( & ( * ( > * * & ( & # ' / & $ % ' & ( * % & ' * $ . % # . * . * / * ( / * $ ' $ *

' . $ % * . / & ( > & ' * & / * ' ( * # % / . * % * * (

( $ ( # . ( * / & ' ( * # * . / * ' / * # $ ' # %

* # . * . & / # # . > & ( # ( > * ' $ % ' * . / $ ( * # . * * #

' # ( $ ( & # ' % * ( / . & ( * ( > * / % # * . $ ( ' # ' ' $ ( % ' * . / & ' /

/ # > $ ( ( > / * * ( $ ( & # ' * # * / ( * # % % # & ' ( > * # . *

Page 185: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � " $ & � & � � � � / 0 � � � �

& � � � & � $ 0 � � � & � 0 � � $ " " 0 � � 0 0 � & $ � 0 � $ �

$ � & � � "

" " 0 � � & � � � $ � � � � " � 0 � � � � � � � � � � � " $ � & �

� & / / � � � � $ 0 � 0 0 � �

0 � � & � � � & � 0 � � & � � � � 0 � � � " �

$ � � & � 0 � & � � � � � / "

& � � � � � 0 & � � " � � � � / � � � / 0 � � $ " 0 0 � & 0 � �

0 � & � $ � $ � " � � 0 " � � � 0 0 � � & � �

/ � � � � 0 � � � 0 � $ 0 � � $ 0 0 � & $ & � � 0 & �

0 � � &

� 0 $ � � / 0 � � �

"

& $ / � 0 � � & � 0 � & �

& � � $ $ / � � � � & � � � $ 0 � � � � $ � � " 0 � & � 0 � � $ 0 0 �

0

0 �

$ " � � 0 � � � 0 � & $ � 0 & � � 0 / / 0 "

� � 0

" 0 � $ 0 � � � $ � � � " " 0 $ � $ � & � 0 � � � / / 0 " � 0 0 � �

� � 0 � 0 � � $ � / $ � � � / 0 � � � � " $ " � / 0 0 � � $ � � � � 0 $ $ �

$ � & � � � � 0 0 � � � � & " �

& � � � � & � & � � $ & � 0 � � � & 0 � � � " 0 � $ � � � �

& � 0 � �

& � 0 � � � $ �

& & � � & � � � $ � � � " � � " � � � � 0 / � � 0 �

� � " � " 0 � � � � $ � " � / 0 � � " � 0 � & � 0 0 $ � / / � �

� � � $ � � 0 � 0 � � 0 � � 0 � � $ � � / $ � � $ 0 � � � � 0 � � � � �

Page 186: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � $ � � * + � � $ 0 0 $ 3 + � * � �

0 �

A � 3 � � A 0 $ � � � 0 � � 0 0 $ � 3 � � A � 3 � � � � � �

� A � * � * $ � $ � 3 � 0 � 0 � � � 0 � � * � � * � $ + * � � $ �

$ $ 3 + � * � � � � � � 3 � * 3 + 3 � 0 0 � 3 3 �

$ * � * � 0 � � 3 � * $ 0 � 0 + � 0 � 0

$ � � 0 � � 3 � + 0 � 0 $ � � 0 � 0 $ 3 � 3 � 3 � � 0

A 0 A 0 3 � � � * � * � $ � � $ 3 � 3 * � * 0 $ * 0

� � A 0 A A � +

� * � A 3 � � A + $ � * 0 + $ � 3 � A 3 � + 3 � + +

� � * $ � A � 3 � 3 � A � � � *

$ 3 � � A � + � 3 0 � � � $ 3 � $ �

A 3 0 $ 0 � � � � 3 + + � � 0 � � 0

� $ � � + � 3 0 0 0 � + � 0

� 0 3 + � � 0 � � 0 $ 0 0

3 � � � � � * $ * 0 �

� � $ � � � � � � 3 $ � + 0 0 � 0 � � 0 �

� � + � $ � 3 � � 0 $ 3 + � * 0 * � � �

3 � $ � 3 0 � � * � � � � * 0 $ * 0

0 *

0

0 $ 0 $ � � * 0 $ * A 0 $

3 0 � � * � $ � � $ � � 0 � $ 0 $ 3 � � * � * � � * * � �

� � 0 � 0 � * � 0 $ 0 � * $ � � * 0 $ * 0

3 0 0 � $ � � * A 3 � $ 3 0 + 3 * + �

* 3 0 * � * � � $ * 0 � � 3 � � $ 0 3 0 3 $ $ � 0 � � � � � �

� � * �

Page 187: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � % & � � & � � & & � � � � % & % 3 � � 3

& ? � % & ? & � % B � & 3 & 3 � & ?

& � & 3 & & ? � � � 3 & � � ? � & B 3 & � � ? � � & � % & �

� % ? � & & 3 � & % & % & B & � � % � & B � % & % & 3 & ? � ?

� & � & ? & � 3 3 � 3 3 & � � ? � � & % & � � � & � % & ?

� � � & % � � � & � & B B � ? � � B & & � ? � & B � ? & � % & �

& B ? % � 3 � % � & � � � � � 3 � % & � � % � � & & ? � %

� � � & B � & % � B & � ? � 3 � & � ? � � � & � % & % % & �

3 3 & � � � � & ? � �

& � & 3 3 & � � & & 3 � B � � � � � &

3 � � � � � & � � & % � %

� & � � & % & 3 B � � B � 3 % � & % 3 & 3 ? ? & � ? & �

& & � � & � B & � � & � � ? � % & � � % � 3 � & ?

B &

? � ? & �

? & & � � � 3 ? & � � � & & � � B 3 � � & � & ? � � ? 3 & �

? & � % ? & � & % & 3 & B ? � � & � � & & ? & � � ?

� B B � B � ? � & ? & � � � & & 3 & 3 % � � & � � ? � � �

� ? 3 ? ? � � � � ? & � � � � 3 � ? � � B &

� & � � � ? % & � ? � B B ? & � � % % & 3 & � � � � � �

� � ? B & & �

? B � � ? % & � � ? � B & ?

� & ? � � � � ? B & � 3 � ? � � & � � � � � � � & � B 3

� ? � � B &

� & � � � � � & � B � ? � % & % & 3 & � 3

& � ? � & & & ? � � � � � � & 3 B � % &

& & ? � % & � ? � & & � � � � � � � � � & % � % & %

� � � �

Page 188: Goldrei.classic Set Theory for Guided Independent Study

� � ! � $ ' ! ! ( * ! ' ! / $ * ( � $ � * ' 1 1 ' *

/ $

� / * � 1 ' ! / $ � / � * � 1 '

/ / $ * ' ( � �

/ $

� / * ( � 1 ' ! ! / $ / ( / � * � * 1 '

$ ' ( � (

( * $ ' $ � 1 / 1 � / $ ' ' ! ! * * ! ( ' 1 ' ' � ( �

$ ( � 1 * � $ ' ! ! * ( * / $ / � ( � / ( ' $ ( � / !

! ' ( ' * ( * ' ( � ( / ( � * � ( 1 * ( / ' ' ! � / / $ ' ( � ( � 1 1

� 1 ' � $ ! 1 $ ! * � / * ' 1 *

� ' ! ' ( ( ' * ' * ' ! ( ' ' ! ! ' $ � � * ! '

( � / * / * � * � � 1 1 ' / $ $ ! � 1 ' ! � $ 1 / $ � ! 1 ' ! ! * ( *

$ $ ! * ' ' ' ! ! ' $ ' / * � ' $ ' ! ! * ' ! *

$ � * � ( / ' !

$ � 1 ' � ' ! / / $ � ' ! / $ � / � ' ! * ! ' $ / $ � $

1 / � ( / ' $ � 1 ' ! � ' 1 � � ( ' * ' ! ( � 1 / ( 1 /

( ( � ! !

/ $ ( ' � ' !

Page 189: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � & ' � � � � � �

� �

� � � � � � � � � � � A � & � � � A � ' � � � A '

� & & � � � � � � � � � � � � � � � � � � & � � � � � �

� � � � � A � ' & � � � � ' � � & � & � � �

� � � ' & � � � � � � & � � � � � � � & � � � � � & � & � & � ' �

� � � � � � � � � & � � � &

� � � � & � � � � & � � � � � ' � �

� � & ' � � ' � � � � � � � ' & � � � � �

� � � � � � � � � � ' � � � � � � ' � &

� & � ' � � � � � � � & � ' � � �

� � � � � � � � � � � � � & � � � � � ' �

� � � � � � � � � � � � � ' � � �

� � ' ' � � � � � � � � � �

� � � A � � � � � � & � � � '

� ' � ' � �

&

& &

& & '

� � � � � � � � � � & & � ' � � � � ' ' � ' �

� � � � � � � � � � & � � �

� � � � � � � � & � � � & � � & �

� � & � � � � � � � � � &

� � � ' � & ' � � � � �

� A � & � � �

& &

� � & & '

� � � � & A � � � � � ' � � � � & ' ' & � & � � & �

� � & � � � & & � � � � ' � ' � � � � � � �

� � � ' � � � � � & � � & ' � � � � � ' �

� � � � & ' � � � � ' � & � � � � � �

� � � ' � � & � � ' & & ' � � � � & ' � �

Page 190: Goldrei.classic Set Theory for Guided Independent Study

! " ! % * % . * 2 * * " < = " = * C " " * % "

% ! % ! < 2 < * " = * " =

* % % = < * O < O < 2 % * % < . % * " = < * " < =

* " " ! * " * ! % < 2 < * " = . ! * " = < = = " < * % * % = < * <

. " O < < 2 < * 2 C <

" O " " O ! < O . " % % < " < = < < % 2 2

% ! . % % ! < . O * 2 * " . * " = < % * 2 2 . 2 " C " * 2 "

< O 2 % = < "

* C . C % ! " ! . " * C % < " < * % % " = * *

= 2 < % " = * % . C . 2 = " < C < " * < C "

. " % % * = 2 *

" 2 2

! * ! * O " ! < " = < % C

. < " = < C " " = ! *

< % ! " < C < < 2

* C 2 < . % % O " * " "

. % ! " % < < 2 2

. < " <=

Page 191: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � ' ( � � � � � � � �

� � � � � � : ( ( � � � � ' � : : � � � � � � � � ' �

� � � � � : � ( � � ( � � � � � � � � � � � ( � � �

� � � � ' ' � � � ( � � � � � � � � � � � ' � � � � � �

� � � � � � � � � � � � � �

( �

� : � � � � � � � � � � �

� � � � � � � � � � � � � � � � ( � �

� � � � � � � � � � � � � � � � � � � �

� ' � � � �

(

( � � � � � � �

� � ( � � � � � � � � �

� : � ( � ' � � � � � � � � � � ( � �

� � � � � � �

� � � � � � � � ( � � � �

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Page 192: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � ! " # ! # � # � # � � # � ! � � � " #

5 � " # � � ! � � 8 � � # 5 � " # " # 8 � 8 # " 8 � # �

� � � � 8 � � ! 5 " � # � " � # � ! # �

� # � 5 � � ! � ! � � " � ! � � � � � ! � " � � ! 5 � " # �

# � ! ! # � !

� ! � � � � # � � " 5 � � � � 5 ! ! � � �

# ! � ! � � # 5 � � ! � # � � � � � � � � � " � � 8 � " #

# # � � # 5 � 5

� � � � � # � " " � � � # 8 � # � 5 ! # 5 ! � " � � �

� � "

# � � � � � � � " � " � � � � � � �

� � ! � � " 8 � � � � � � 5 � � " � 8 � � " 8 �

# # # 5 5 � ! # � � � # � � � # � � " 8 � # # �

� � � � � � 5 � 5 � � � � # � � � � ! � 5 � � � � " 8 � �

5 � � � # 5 5 � � � " � � � " � � � �

8 " � � � � � " # � � � # � � � � � � � � � � � " � ! � !

5 � � 8 � � � ! # 8 � � � � 5 5 � � �

" � # 5 � � ! # 8 � � � " #

8 � � � ! # � � � " 8 � ! � # 5 � � ! � � � � � � � � � # � "

8 � � � " 8 � 5 � � 5 � � � � � " � � � 5 � � � � #

� " 8 � " 8 5 � � # � � � " � � � � � " � ! � � � � # 5 �

"

8 � 5 � � � � � � � � � � � � # � � " " " 8 � 5 � � # � ! � # " 8 � 8 � �

# � " # � ! � " � � � � # 5 # � � # � � � � " � � # # ! " �

� 5 ! # " � � � " 8 � � � � � " # � ! � # � � " 8 �

� ! � � # # � �

Page 193: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � �

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Page 194: Goldrei.classic Set Theory for Guided Independent Study

! # ! $ % & & * . ! ! $ ! C * % # $ ! # G *

$ % ! *

! & * C W . ! & * & ! G # & # & G $ . W & $ & # $ &

! C C # . . ! # # ! $ & . * $ % $ $

G & # . % $ ! # C & ! * . C & ! & G $

$ % $ ! $ W . & % * . $ ! . ! $ C & # % . W & $ $ % & % & * $ G *

W & & . * ! % C & % ! . & $ ! ! & . $ ! & W & & ! $ & * # ! $ %

& % & & $ $ % # # % . W C & & * * $ $ # W C & . * W % . . W $ $ & %

& ! & $ % & & & W *

# C $ C # W

* . W # * C & # & C

W * & $ G & # & W C & # W & $ & W * # ! $ G ! & # % $ % & % # !

$ $ & ! # & ! $ % & & . G . & $ % & % & * & W * ! # & & G G $ % & & & W * . !

# W # % $ % # G . G ! $ & & # G & ! & * & $ & # % &

& G G $ & & * & W ! & & G G W G ! & % * # ! W & & W # G G

! & C # G & ! & # ! # W C & $ ! $ % ! # G * ! # * # ! $ W

C C # % & . & & ! % $ & $ % & C & $ & W . $ & # . $ ! * $ C C W & W

C & W $ ! & $ % . ! G $ & # $ ! G G & & * $ % & % . * $ $ C .

$ W C $ * & $ W # . ! & ! W C & $ & % W C & $ % & W $ & % # W $ ! *

& W & G ! & % G $ % & & * & W & W W # ! $ % C & $ & % & * & W

$ % & $ W C & $ ! $ % & % & * &

$ & % & & $ ! W . W C C & * # G $ % &

. . & & # $ & % $ ! C & & * $ C W W C W C $ % & % . &

$ $

#

Page 195: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � ) * � , � � � ) � � � � ) � , � � � 7

9 , 9 � ) � * � � , , � * , � * 9 � � � 9 � � � � � � ) � � � 7 * 9 �

* 9 � ) � * 7 9 7 7 � * , � 9 � 7 � � � ) � � 7 � � � , � � * � � * , � �

� * � � * � , ) � ) ,

� * ) � , � * 9 � 7 7 � � � � � � � � , � 9 � , � �

� � ) , * 9 � 7 * � 7 � � 7 7 7 * * 9 , � � * 9 � � � *

9 � � ) � � � * * 9 7 � � � 7 9 � � � � * 7 � � � ) � 7 �

* 9 � � � ) � * 9 � * � � 7 � � � � � 7 * � � � � � 7 7

9 � � � 7 * � * 9 � � 7 * � , � , 7 9 � , � � � � ,

7 � * 9 � , � � � , � 9 , 7 � �

* * 9 , 7 � * � 7 � * � � � � � * � � � � � * 7 � , � ) , � �

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, 7 � � � � * � * � � � * � , � 7 � � � 7 � � � � � *

, � 9 � ) � � � � � � * 7 * 9 � � �

� � 7 � � � � � � � ) � 7 � , 7 , � * 9 � � * 9 � 7 � �

� � � 9 � * 9 � � � 9 � � � � 7 , � * 9 � � � * � � ) , 7 �

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, � , � � ) �

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9 9 � * ) � )

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7 9 7 � � * � � � � * � � � � � � 7 7 � � � � � � * , 7

� � � �

, 7 � * � � ) � 7 � * 9 � 7 7 � * � � 9 � � � � 7 * � � � � �

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� 7 � � � 9 � � ) � ) � 7 � 9 7 � � � , � 9 � � *

� � � ) �

� * � � � , � 7 * � � � � 7 7 � * , � � � � � * �

* � � � � 7 � � � * , , � 7 � � * � �

Page 196: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � # � � � � � � � � � * � # � � � � � � � �

� # �

* � # # � � � # � � � � � � � � � � # � � � � � � � � * � � � #

* * � � � # � � � � * � # � � � � � � � � � �

� � � � � � � # � # # � � * * � � � � � � � � � � �

� * � # � � # � � � � � � � # � � � � � � � � � * � �

� � � � � � � � � � � � � � � � � � # * � � # � * � � � � # * � � � � � � � � � �

� � � � � � � � � � � � # # � � � # * � � * � � � � � � � # �

� � # � � � � � � � # # * � * # � � � � � � � # * � � � � # �

� � � � * � � * � � � � � # � � � � � � � � � * � * � � � �

# � � # � � *

� � �

� � � �

* # * � � � # � � � � � � � * � # � � * � � * � � � � # � � � � � � � � * �

� � � � � � # � � � � # � � � � � * � � � � � � � # � � �

� � � �

# � # � � � � � # � � � � � # � * � � � � � � � � � �

# � * � � � # � � � � � � � � # * � * � # � � � � �

# � �

� # � � � � � � # # � � � � * # * # � � � # � � � � � � � # �

* � � � * � � � # � � � � � � � � � � � � * � � �

� # � � � � * � � � # � � � � � � �

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� � * � � � � � # � � � * � � � � � � # � � � � � # * � � � * � � � � * � � �

� � � � � � � � � � � # # � � � � � * � # # � � � � � � �

� � � � � � � * � � * � � � � � � � � � � � � � # � # � # � � �

� � � � � � � # � * � � � � � � � � � � � � # � � � � � � * � � � � � �

� � � � � � � � � � � � * � � � � �

� # � � � � # � � * # # � # � � � � �

� � #

� � � � � � # � � # � � � � * � � � � � � � � # � � � # � � � � * � � � #

* � * � � � � � � � � � � � � � � � # � � # � � � � � � � � � � � � � � � * �

� � # � � � � �

� # � * � � # � � � � � � � � � � � # # � � � �

# # � � � � �

Page 197: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � % � % � . � / . 1 / . . � � � 1 � . . � / :

� � . � . � � . � 1 � �

� � � � / � 1 . � � % � % �

� � / � . % � � % � % � 1 � 1 / . � � / � : 1 / . � %

� � � � � / � 1 � } � % � � � � 1 � 1

1 � / � . % � � } � � % � � � � � � / � � � � � � % � �

/ % � � � } � � �

� � } � � % � � � 1 � % � � � : . % / � % � : � � � %

� � / / � � � . : . � � . � � . � � . / / } . � % 1 � � .

� � } � � � � : : } � � . � % . � � % � � � � � � � � � � �

/ � } � � % � . � � � : 1 / % � � . � � � � � . � / � � �

� : � . � . � � � � � � %

� � � � / : 1 � � � � � � � � � � � . � / � � : � . � �

� � � %

1 � �

� } � . � 1 1 � �

1 . � � � � � � / 1 � � } � � . �

� / � � � � � � } � % � % � � � � / } � / � / . � 1 /

� � � � � � } . / � � � : . � � 1 : � � : �

� � � % � � 1 / } � � } � � � � � � � . � � � � .

. � � . � � } } � % � : . � � � } . � � / . � 1 / % � } � 1

. 1 � � � � . . � . � / � . : . � � � }

� � / 1 � � 1 1 1 � � � � . � % � 1 : � . /

� } } % � % � 1 � � � 1 � / � . / 1 � � � � � % � 1 : � �

� � � � � 1 � �

� � � / 1 � � � � % � % � � � � � � � � /

� � / � � � � % � � / � � � 1 } � � / .

� � . � � � � � . � / � . � � � � / � / � � � .

� � . 1 � � % � % � � / } / � / � . � � � /

/ � / � . � � � � / .

Page 198: Goldrei.classic Set Theory for Guided Independent Study

� � � $ � ' � � + ' , $ � ' � � , � 4 � � � ' �

J J � $ � $ � + � O ' � , � � J � $ + � � � � � , � $ '

� , � ' � � O � 4 � $ + � ' � O $ , 4 � ' $ J + � ' � , � � �

, � $ J , � � � � ' ' , � � � ' ' � � J � ' + � � � O � +

� , � , � � 4 � ' $ � J � + � 4 , � � $ ' � , � � � J � � '

� + � , 4 � � ' � � ' � ' 4 ' $ , � � O O � � ' � J

� � � O $ , , � � $ , � $ + + � � $ + O � �

� $ , � � + � , + ' ' � , � ' � , � � , � � J � , � J � � ' �

� � � �

, � O + � � J , '

, � � O 4 ' � � , ' � , � � + � 4 � 4 +

� , � $ + $ ' � O � � � ' � � � $ �

+ � $ , � + , ' � ' 4 4 � ' � � ' + $ '

� � $ �

$ + � 4 �

4 + � � � � + , � � � � � $ ' � � , � ,

� , �

' � � , � , � + � , 4 � ' � + � � O � $ � + J , � , , � � � $ 4 + � ' �

� J J � O � + + 4 , � � $ � � � � 4 � � � � O � ' � � � � O � + � +

� ' � � � � $ J � ' , � � � , � � O J 4 ' � $ J � O � � ,

' ' � � + + 4 J � J � � O � � ' , � 4 ' $ J $ + , � , $ � �

4 ' , $ ' � ' � , � � � + $ J

$ ' � , � � � $ ' ' � O � O � � 4 '

� � � ' � � � � 4 '

$ � � + � 4 + � J J � O � O � + � � � + ' , � ' , $ ' $ '

' � � � ' � 4 ' � � O � 4 + ' � � � O � � ' � �

� � ' � � + ' , � � � � ,

� � � 4 , � � � � � � � $ ' � � O � � J � ' J � , �

+ + � $ � � � � ' , � � O , � � ' ' , � J $ '

� � � � � ' � , � � � , � � ' ' � � � O O � + � 4

4 + ' � � � , � $ � J 4 ' � � O � O � , � � O �

� � ' � � O � � ' � � � � ' $ $ ' , � �

� � � + � �

Page 199: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � � '

� ' � ' � � � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � !

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� � � � � � ! � � � � � � � � � ' � � � ' � � � ' � � � � ' � �

� � � � � � ' � � � � � � � � � � � � � � � � � � � ' � � �

� � � � � � � ! � ' � � � � � � � � � � � � � � � � '

� � � � � � � � � � � � � � � � � ! ' ' ' � � ! � � � � � �

� � � � � � � � � � � ' � � � � � � � � � � � � � � � � � � � �

! � � � � ! � � ' � � � � � � � � � � ' � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � ' � � � � ' � � � � � � ! �

� ' � � � ! ' � � � � � � �

� � � � ' � � � � � �

� '

� � � � � � � � � �

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� ! � ! � � � � � � � � '

� � � � � � ' � � ' � ' � � � � � ! ' � !

� �

� � � � � � � � � � � � ' � � � � � ! ' � � ' �

� � � � � � � � � ' � � � � � � � ! ' � � � ! � � � �

� � � ' � � � � � � � � � � � � � � ' � � ! � � � � � �

� � � � � � ' � � � � � � ' � � � � � ' � � � � � � � � � � �

� � � ' � � � � � � ! � � ' � � � ! ' ! � � � ' � � !

� � � � � � � � � � � � � � � � ' � � � � � ' � � � ' � '

Page 200: Goldrei.classic Set Theory for Guided Independent Study

� " % � � ( � ) + ) 2 � 3 ) 2 � %

� � D D + ) ) � 3 K ( � + ( � ( � )

% � � K % � K ( % 3 % 2 � 3 ( � � 3 � %

3 % + 3 3 � 2

� 3 3 % ( K ) � � K 3 2 � % D � K % ) % K + 3 � % � D � � ) (

� ) K 3 � � " % D � � ( � ) � � K ) % � % ) � % " " % 3

2 % (

( % � D % % D 2 + ) % 3 + + " ( � � 3 %

( % � ( % 2 + ) %

+ % � % D 2 + ) % 3 + ) " ( � � 3 %

� ) � 3 � K 3 % ) 2 � + ( "

% D � � % 3 ( % D � D ( % % 2 + ) % K 3 + 3 + + K ( � � 3 ) � %

) 2 � % 2 � D � � 2 + 2 ( � ) + 2 + 2 3 � ) + 2 � � " � � 3

% � % % �

% + % � � + � 3 3 + 3 + ) � � " " % � D � � K � ) � D

2 � K � % � D % % 2 + % % 3 � % � D + ( % % D + % K 3 + 3 + ) + K � � ) %

2 � % 2 � % � ) 2 K ) 2 � � 3 % % � + ( % % D 2 � + � 2 + 2 3 � +

% � D ( % + ) % 3 3 + ) + " � � 3 ) % ) � % ) 2 � D 3 + � � % � ( % 3 + 3

% D ( 2 + (

( ( ( 2 K ( + + 3 3 + � ( � + ( � � " + � � ( ) K + ) ( K ( % �

+ ( % 2 + % + 3 + + " � � 3 ) K ) 2 � 2 3 � % 2 � K D % K 3

2 + ( + ( + � � ) 2 � K ( � � ) � K ( � ) 3 % + 3 K 3 � 3 � 3 + % 3

+ 3 K 3 � + 3 3 �

% ) 2 � D + �

Page 201: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � % � � � & �

� � � � � & � � & % � � � � & % � � % � % � & � � �

% � � & � � � � � & � � � % � � �

� � � � � � & � � & % � � � �

� & � � � � � & � � � % � & � � � � � � & � � � � � � � �

� � � % & � � � � & � � � � � � � & � & � � �

� % & � � � � � � � % � & � � � � � � �

� � � & � � � % � & � � � � � % � � � % � � � � � � % � % �

� � � � �

� � � � � � % � % � � � � & � � � � % � � � � � � � �

% � � & � % � % � � � � � � � � � � � & � � � � � � � � � & �

% � � & � % � � & � � � � � � � � � � � � � � � & � � � % � � � � � �

� � � � & � % & � �

� � � % � & � � � � � � & � & � � � � � � � � � � � � & � �

� � � � � � % & � � � � � � � � � � � � � � � % � � � � � � � � � � �

& � � � � � � � � � � � � � � � & � � � & � � � & % � � � � � � � � � � �

� � � � � � % � � � � � � � � & � � � % � � � � � � � �

� � � � � � � � � % � � � � � � % � � � � � � � �

� � % � � � � � & � � � � � � � � � � � � � � %

� � � & � � � � � % & � � � � � � � � � � % � � � � � % � � � � % � �

� � � � � � � � � � % � � � � & � � � � � � � � % � � � � �

� � � % � � � % � � � % & � % � % � � & � � � & � � � � � � �

� � � � & � % % % � � � � � �

� � � & � � � � � � � � � � � � � � � � � � � � % �

& � � � � � %

% � � � & � � � � �

� � �

� � � � � � � % � � % � � � � � �

� � � � � �

� � � & � � � & � � � � � � % � � � � � � � � � � � �

� � � � � � � � � � % � � � � � % � % � � � � � � � & � � � � �

� � � � � % � �

� � � � % � � � � % � % % & � � � � � & % � � � � & % � � � � & �

� & � � � � � � � � � � � � � � � & � � � & � � % � �

� � � � � % � �

Page 202: Goldrei.classic Set Theory for Guided Independent Study

. . 2 3 3 . : . . = ? . . 3 . 3 : F : 2

: . F = . . e e

? . . . e

:

? F : ¡ e ? : . 3 . F ? ? 3 F e 2 = . = . . ¡ : 2 « 2

. : . : . e F . . 2 . . 3 ¡ :

. . . . . 3 3 : 2 e : « ? . F F : . e ? ? . .

F 3 2 3 : 2 F . . . . . e ? : . « F

. F 3 e ? . : . ? « .

. . 3 3 . F F ¡ 3 ¡ ¡ 2 e ¡ ¡ ? 3 . F : F ? . 3 e ? F . 2

. e : . 3 3 . F ? ¡ 3 ? . F ? ¡ 3 . = = ? « . « . 3

2 = . 2 . = . . 3 2 ¡ 2 e e ? F . F : . . 2 3 . = . . F ? 3 ? F : 2 F

. e . F : . ¡ ? F : ¡ ? « 2 e . F

: 2 . ¡ . : ¡ . e : 2 F . 3 ? : 2 ¡

? F « 2 ¡ ¡ e F : 2 F e

:

: . . : ? . ¡ e : ¡ ? ? . 3 « . 2 3 F : 2 F ? F : . ¡ e ¡

? . . = . 2 3 . : 2 F : . ? : F ? . 3 = ? . 3 e ? F : ? = ? « 2 = : .

. = . . e 3 ? 3 : F 2 3 ¡ . = = ? « . . F : . « . 2 = . 2 F . = . . F ? ? «

: ¡ : ¡ ? . e F : = F . . ? : 3 = . F . = . . e F ¡ e 3 ? . 2 ? ? F : . ?

« F ? ? F 2 e ? e F . « . = .

. 3 : 2 = . F : . e . F « . 3 = F . . . ¡ e ? « « ? ? ? : . ? « . e

¡ ? F F : « ? ? 2 = 3 ? . ? e . F : . . F : ? ? F : . ¡ e ¡ 2

. « . = . : . : = = ¡ . ? 3 2 e . . ¡ .

2 e ? . . . e ? 2 . = = ? . « . 3 . F ? ¡ . =

: . e ? « 2 = =

. e . F : . 3 . F ?

. 3 : 2 = 3 : ? e ? e : . ? « . F : 2

3 . : 2 F 3 e ? . F 3 ? F 2 2 3 2 = . 2 3 F . = . . e F : . e 3

: 3 F ? . . . = . . ? . : 2 . : . ¡ e 2 = ?

: : . = . F . = . . ? ¡ ¡ e

Page 203: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � �

� � � � � � � 9 � > �

� � � 9 � � � � � �

9 � �

9 � > 9 � � � � � � >

� � � � 9 �

� � � � � � 9 � � � � � � ^ � � � � � 9

9 � � � � � � � � � � �

� � � � � � �

� � � � � ^ 9 � ^ > � � � � � � � � � � � 9 ^ � � � � � 9 ^ � > �

9 � � � � � � � � � � ^ � � � � � � > � � � � � 9 �

� � 9 � � � � � 9 ^ � 9 � � ^ � � � � � � � � � � � � �

� > � � � 9 ^ � � ^ � � � > � � � � � � � � � � � � � 9 � � � � � � � > �

� � � � � �

� � � 9 � � � � � � � � � � ^ ^ � � � � � > �

� � ^ � � � > � � � � � � �

9 � � � 9 � > � 9 � � � � � � > � � � � � � > � � � � � > � �

� 9 ^ � � 9 � � � � � � � > � � � � � 9 � ^ � 9 > � � � � � � ^ � � � �

9 ^ � � �

� � � ^ � � � 9 � � � � > � � �

� � � � � � � � � � > � > � � � � 9 � � � 9 � ^ � � � � � � � �

> � � � � � � ^ � � � �

� � � � � 9 ^ � � ^ � � � � � 9 � � � � � > � � � � � � �

� � � � � � � � � � > �

Page 204: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � $ � � $ � � � � � � � � � � � � � � �

� � � � � � � � � � � � � $ � �

� � � � � � � � � � � � � � $ � � � � � $ � �

� � � � � � � � � � � � � � � � � � �

� � � � � $ � � � $ � � $ � � � � � � � � � � � � � $ � $ � � $ � � � �

� � � � � � � � � � � � � � � � � � � $ � � � � � � � $ � � � $ � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � $ � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � $ � �

� � � � �

� � � � � � � � � � $ � � � � $ � �

� � �

� � � � $ � � $ � $ � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � �

� � � � � � � � � � $ � $ � $ � � � � � � �

� � � � $ � � � � � � �

$ � � � $ � � � $ � � � � � � � �

� � � � � � � � � � � � � � � $ � � � $ � � � � � � � � � � � � $ � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

� �

� $ � � � � � � � � � � $ � � � � � � � � � � $ � � � � � � � � � � �

� � � � � � $ � � � � � $ � � � � � � � $ � � � � $ � � � �

� � � � � �

� $ � � � � � � � � � $ � � � � �

� � � � � � $ � � � $ � $ � � � � � � �

� �

� � � $ � � $ $ � � � � � � �

� $ � � � � � � � $ � � � � $ � � $ � � � � � $ � $ � � � � � � � �

Page 205: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � �

� � � � � ; � � ; � � � � � > � � � � ; � � � �

� � � � � � � � � � � � ; � � � � � � �

� � � ; � � > � � � �

� �

� �

� � � � �

� �

� > > � � � � > � � � � � � � � � � ; � ; � �

� � � � � � � � � � ; � � �

� > � ; � > � ; � � � � � � � � � � � � � � � ;

� ; � � � � � � ; � ; �

� � � � � � ; � � � � � � � � � � � � � �

� � � � � � � � � � � � �

� � �

� �

� � � � � ; � � � � � � � � � � � � � � � ;

� � � � � > � � � � �

� �

� � � � � � � � � � > � � � �

� �

� � ; � � � � � � ; � � � ; � � ; � ;

� � � � � � ; � � � � ; � � � � � � ; � � � � � � � �

� � � ;

� � ; � � � � � ; � ; � � � ; � � � � � � � � � � � �

� � � � � � � � ; � �

� � � � � > � � � � � � > � � ; � � � � � � � � ; � ; � �

� > � � � � � > � ; � � � � � � � ; � � � � � � � � � ; � �

� ; ; � � � ; � ; � � � � > � � > � � ; � ;

� � � � � > � � � � � � � � � � � � � �

� �

� �

� ;

� �

Page 206: Goldrei.classic Set Theory for Guided Independent Study

� � ! " ! � �

� � � � " 3 4 6 3 6 " 3 � � " 6 3 6 � " ; � ! " @ ;

� 3 @ � 4 � � 4 ! " � � 4 � 3 @

� 3 @ � " ! " � 3 @ � " 6 @ � 3 4 3 @ � 4 ! 6 ! "

� 4 � 4 6 ; � � 4 6 ; 4 � �

� � 3 @ ! 3 � " 4 � 4 � 3

! " � 6 ; 6 " 6 3 6 � � 6 y

� ! y �

� 6 � " ! � ; 3 � 3 6 � �

� � 4 @ 6 3 � ! " " 6 3 6 ! y � " 3

6 6 y ! 4 y 6 ! " � " � 4 � � ; ; � � @ � " 6 ;

� � 6 ; � � 4 6 � " 6 3 6 ; � � �

� @ ! � ! y ; 3 � y � � � � � � " 3

� � � @ � � y y � 6 " � y

6 ; � � 4 6 � � 4 6 � " 6 y

6 ; 4 � 4 6 � � 4 6

6 ; 4 � 4 6 � � 4 6 3 � " " 3 y �

� ; " � 3 3 y � � � �

@ y ;

� y � � 3 � � � � � � � 6 " 3 � 3 � " � �

! � � " � y

! � � �

� 3 ! 3 � @ " ! y y 4 � � y y 6 y �

4 � 6 3 4 � y 3

� ! 4 y � 4 � 6 � � 6 3 " � 6 y y 6 � � " y y � 4 3

" � 3 " 3 � y 3 � � 6 y " y � 4 � y � 6 �

� � 3 ; 3 6 � ! " " " 6 � � y ! " � " 6

� � 4 ; 6 y y 6 � � y � ; � y y � ; ; 6 y y y � 4 � �

! " 3 � " � 4 � 4 � ; 6 � � 4 6 � � 4 6 6 " 6 y � � "

� 6 3 � y " � 3 6 6 � � y y � 6

" � 3 ! � 4 ; @ y y " 6 y y �

Page 207: Goldrei.classic Set Theory for Guided Independent Study

$ % ( % * , * , $ . / ( , 1 . . , 1 * 1 , . 1 6 , , , ; 1 , 6 % ; 1 1 ( % 1

$ , * , 1 6 , 6 , , / ; , 6 ; * 1 ; , / ; / ( , 6 , $ * , . 1 $

, . 1 , , , / 1 ( % ( / ( ; 1 . , % 1 * % ; * , , ; ,

, / , . 1 , , ; , ( 1 / ( / 1 , % ; 1 1 6 ( / 1 ( , ; ,

, 1 . % / % $ . ; , , 1 ( , 6 , / ( , , $ , 1 6 , ,

, / , $ ; , / 1 , . , 6 , , / ;

$ ( 1 / ( $ / 1 * / % * ; $ , % / ( 1 , 6 / ( $ 1 , $

, % ( 6 6 % ; 1 ; 1 1 ( % 1 * % * % / % $ . ; , , / 1

1 , ; 1 / , , % / ( / ( , 1 , / ( / 1 6 , $ (

, % ( , 6 % ; 1 6 , 6 % ; 1 1 ( % / 1 * % / % $ . ; , , * / 1

, / , . % * $ . 1 , , ; * , ; , 1 , 6 , , , / % , . $ ; , 1

( 1 ( $ / ; , . . 1 6 , , ( , * / ( , , % * 1 % * * / , , ; , * * ( * 1

, . , , * / ; 1 % , / ( , 6 , % ; * 1 ; , , * , * ; 1 ; ( $ /

; , 1 * , ; , 1 / ( ; 6 , ( 1 . % , % 1 * $ % * ; * 1 % * * , , ; , * *

( $ * ( , % ; , . 1 6 , 6 ,

1 6 , 1 ( , 1 . 1 % * . % * , $ 6 . 1 , 6 , , / ; , , , ( , 6 ( , 1 , % ;

$ 1 , . 1 6 , 1 6 , ( ; , ( ( , . 6 , 6 1 1 , ; 1 $ 6 % *

/ ( , , * ; , 1 ( , 1 , / ( / ( , , / 1 6

$ ( , ; ; , 1 / / ( , ; . , 6

, ; , / 1 ( ( , ; . 1 6 ,

( , ; , 1 1 % , * / , , ; 1 6 , 6 , , * ,

% , % / ( , 6 ( ; , , % % ; 1 6 ; ( $

1 $ * ( , / ( , ; $ , * , 6 1 % 1 6 ; *

( , 6 , / , $ . 1 6 , 6 1 *

( , , / 1 . , , * , ; 1 * $ / $ . * , % $ 6 , 1 $ 1 /

, , ( , , % , , % * ( , , , $ * / % 1 * 1 / 1

1 6 ; 1 , 1 $ . .

1 6 , , ( , 6 ,

/ , 6 , ; ; 1 , ; / $ /

$ * 1 $ .

Page 208: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � �

� � � � � 9

� � � � � � � 9 9 � � � � � � � � � � � � � � � � � 9 � � � � � � � �

� � � � � � 9 9

� � � � � � � � � 9 � � � �

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� � � � 9 9

� � � � � � � � � 9 � � � �

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� � � � � 9 9

� � 9 9 � � � � � � � � � � 9 � � � � � � � � � � � � � � �

� 9 9 � �

� � � � �

� � � � � � � �

� � � � � � 9 9 � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � �

� � � � � � � � � � � � � � � � � � � 9 9 � � �

� � � � � � � � � � � � � � �

� � � � � � 9 � � � �

� � � � � � � � � � � � � � �

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Page 209: Goldrei.classic Set Theory for Guided Independent Study

� � $ � & , $ , , , , , 2 � & 7 , 9 2 � & ? @ 2 � & � &

? , @ � , ? $ , H ? & 2 & @ , � 9 , , $ , ? , @ �

$ 2 , ? ? @ , , @ � H 7 7 � 2 & ? � , $ , 9 2 ? � & @ , @ 2

H 7 @ � 2 & � 2 , , $ & , 2 � $ � � � H 2 & ? & & 2 @ , 7 H & � 2 & @

� H & @ � & ? 2 , $ , @ , , , & , � ? , @ � H 2 , ? @ H $ 2 & @ 9

2 & @ ? , , , @ @ , & , , 2 , , � H & ? � � &

, H 7 7 @ H H & @ � ? 2 & @ , , , 2 & 9 & 2 , & , ? $ , $ , , & @ H 2 � & �

2 & � & H $ 2 , $ , $ 2 , ? & 2 & � , , & , 9 � H $ � & , 7

? 9 9 , ? , � & 9 $ 2 , $ , 2 , � $ 9 H � , , �

, & ? $ , 9 H @ @ 9 , ? , , ? , $ , & H @ 2 , @ 2 H 7 @ 2 & ? 9 , � 7 , , &

� 2 & 9 , � & 2 & , & � � & 2 & @ , H , 9 , & 2 ,

@ H ? � , & � , @ 9 , , $ 2 , ? , @ ? 2 , @ , ,

9 , H & � $ $ , ? & ? � , 2 & & 2 @ , & , &

H 9 2 , & ? @ 9 H 2 ? 2 & 2 @ , ? � , 9 � @ H , H $ ,

$ @ , & @ , @ 9 H 2 2 & , & � & $ 2 7 � ? , 2 & , $ H 7 9 , &

, H 9 � � 7 H 7 � & 7 , 2 @ � 9 H & � & , � 9 , 7 H @ , $ �

, ,

H & @ � $ ? 9 , � � & 2 , � & @ $ @ 2 � & 2 & , H , , & & � & , �

@ H , @ , 7 2 H , 7 2 � & , � � & @ $ 2 &

9 , 2 & @ $ , @ 2 & , , & H , , , , , � , , 2

, ? $ @ � & � & @ 9 2 ? ? , � 2 , & 2 & @ H , 9 &

& � & , � 9 , ? , , @ 2 & @ , 2 & , ? , @ & @ � 2 ,

& � � &

7 H @ , 7 H � 2 , , , ? $ 2

& @ 9 , 2 � & $ 2 9 @ @ 9 H ? & , , 2 � � , & H @ 7 &

, $ ? , 2 & @ $ � , @ , � @ H @ 2 � & H @ 9 , $ @ 9 H & @ � , ? $ 2 , ,

2 & & 2 , � & , � , & @ , $ � $ , $ , 2 , 2 & � , ? , & ? , H @ ?

2 � & � @ @ 9 , @ H , 9 � & , , ? � & H @ @ , $ , 2 �

$ � , ? ? � @ H 2 & 2 @ ? ? , 7 7 , & H 2 & , $ � 7 7 � $

, @ 9 , , , & � $ 2 & H ? H & � , $ , , 2 & H ? 2 & � � &

H & � , $ � 9 7 , 7 � � , $ 2 � � $ , 7 7 � $ $ , ? , ? ?

@ 9 , � $ , $ , � H 7 � $ , $ , @ $ , $ � � $ 2 @ � 2 @ 9 2 @ 7 � $ $

9 2 � , $ @ , H ? , , @ & H & & 2 , � , $ @ , , 2 9 2

@ , � $ , $ , � H & � , , , & @ 2 , H � , , @

9 2 ? 9 � , , 2 , ? $ 2 , @ � � & 2 , $ H 7 � 7 , 9 , & @ $ 2 & @ � $ , $ , ? , &

H @ 9 , @ 2 H 7 � , 2 & $ ? , @ ? , , , $ @ 9 H @ , H 2 & $ � ,

� � & � @ 9 , $ � $ � ? , , � $ ? $ , , � H @ , H & � $ ? �

2 & & H ? , @ $ , , , & H & � , $ 9 , & H @ $ 7 , @ � @ H , 2 ? ,

� 7 , � H 7 7 $ , $ , @ ? � H , & � $ , , H ? , , � 9 � ,

2 & , $ 2 ? , @ 9 , , & , $ , 2 � 9 H @ $ @ 2 ? 2 , �

Page 210: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

> � � � � � � � � � � � � > � � � � I � �

� � > > I � � � > I � � � � � I � I � � � � � � � � � � � � � � � � I � �

I � � � � � � � � � > � � � I � I � � � > � � � � I � � � � �

� > > � � � I � � � � � � � � I � � � � � � � � � � � �

� � � � � � I � I > � � I � � � � � � � � � � � > � � � � �

� � > > � � � � � � � I � � � � � I � � � > � � � � > � � � � I � � � � � > � � �

� � � � I � � > � > � � � � � � � � � � � � � � � I � � � � � � � I � �

I � � � � > � � � � � � � I � � � � � I � � � � � � � � � � � � � � � � � � � I �

� � � � � � � > � � � � > � > I � � � � � I � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � I � � I � I � � I � � � � � I � � � � � I > � � > �

> � > � � > � � � I � � > � � � � � � I � � � � I � � � � � � � � � � � � � � �

� �

� > � > � � � � � � � � � � � � � � � � � � � > � I � � I � � �

� � � � � � � � � � � � � � I � > I � � � � � � � �

� � � � � > I � � � � � I � > I � � � � � � � � � � I � � � I � �

� I > � � > � � � I I � � � I � � � � � I � � � � � � � > � � � � � �

� � I � � � � � � � � � � � � � � � � � � � � > � I � � � � � � � � > � � � I �

� � � � � � � � � � � � � � I � � � � � � � > � � � � � � � � � � � > � �

I � � � � � � � � � � � > � � � > � � � � � � � � � � � � I � � � � � � � � I

� I � � � � � � � � � � � � � � � � � � � � � I � � � � � � > � � � � � � � � � � � � > �

� � � � � � � � � � � � � � � � > � � � � � � � � � � � � � � �

� � � � � � � � > � � I � � � � � I � � � � � � � � � � � � � � � � � � > � � � � �

� � � � > � � � � I � � � � I � � � � � � � � � � � � � � � � � � � � � � � � �

I � � � � � � � > � � � � � � � � � � � � � � � � � I � � � I � � � � I �

� � � � � � � � � � I � � � � � � � � � � � � � I � � � � � � > � � �

� > � � > � > � � > � � � � � � � � � � � � � � � � � � � � � > � � � �

� � � � � � � � I � � � � � � � � � > � � > � � � � � � � � � � � � � � � �

� � � � � � � � � � � > � � � � � I � � I � � � � � � � > � � � � � � � �

> � � I � � I � � � � > � � � � � � � > � � � � � � � � � � � � I � � � � � I � I �

� � � � � > � > I I � � > � I � � I � � � � � � � � � � � � � � �

> � � � � � � � I � � > I � � � � � � � � � � � � � � � � � � � �

� � � � � > � � � � � � > � � � � � > I � � � � � � � I � � � I � � I �

� � � � � > � � � � > � � � � � � � � � � � � � � � � > � � � � � � � � � I � � �

� I � � � � � � � � � � � � � � � � � � � I > � � I � � � � � � � � � � � � � � � � � �

� � � � � I � � � � I � > � � � � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � I � � � � � � � � > � � � � �

� � � � � � I � � � � > � � � � � � � � I � � � � � � � � � I � � � � � � � � �

� � � � � � � � � � I � � � � I � � I � � � � � � � � � � � � � � � � � � � � � I � �

� � � � � � � � � � � � > � � � � � � I � � � � � � � � � � � � � � > I � � �

� � � � I � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � �

Page 211: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � ! � & � � ( ( � � � � � ! � * * & * ! � � * � �

! � � � ! � � � * � ! & � & ( ! � � � * � � � ! � � * � ! � � � � *

* �

O & * � � ! & � � � � ! � � O * � � * ( ! � � � * � ( � � ( � � �

� & � ! * � * � ( � � � * ! �

* ( ! ( � * � � � * * � * � & ! & � � * & ( ! � � � * �

� & � � � ( & � ( ! � * O & * * � * ( � ! � � � � � & ( *

� O & � �

� � � � � � � * � O ! * � & ( ( � � � � ! � O ( & ! ! � � ( ( � � � �

� * � � * ! � � � & ! � � � � � � & * � � � & � � � � � � �

* O � ! � ! � � � � ! � � � O * � � O & & ! � ! & � * ! � � ! ! & ( (

� � ( � * * � � * � � � ( & ! ! � � � � ! � � * � � ( ( � � � � � ! � * � * � ( (

� � � O & � � & ( � O � O � � � ( & * � � ! * & ( ( � ! � * � � � ! O !

� � � � * O � � � * & ( ! * & * � � � * ! � * O � � � ! � � * O ! & * ! �

� ! � & * O � � � � � � ( * � � � ! � � � � � * � O & � � & O � * O � ! � ! �

& � � � � � O � ! � ! ( ! � & � * � � � ( & * � � � ! O � & �

! � � * � � & ! � ( ( � � � � � ! � * ! O � ! � & ! � � � * O � � � & * � !

� � � � � � � * O � ! � & ( � � � � � & * � � ( ( � � � � � * ! & � ( * O �

! � ! & * � � �

� � � � � � �

O � �

( ( & * ! � ( & � � � � & ( ! � � � � � � & & � * � � � ! � � * O !

� O � � � & � � * O � � � & (

& � O � * & * � * O & � � * O � � � � �

� � � � ( � � * � ! � � ( ( � � � � ! � * ! � * � � * O & � * O � & � �

� � * � � & � * ! � ! � � � � !

� & * ! � � � & (

! � * � � � � �

& O � � ! & � � � & ( & ( ( * O � � ( � � � * ! � & � � ( � � � * �

� & � * & � � & � O O � � * * � � � � �

� O & � * O & * & ! ! � * � * � � � ( � ( � � * �

O ! ! � * & ( * � � * O � � � � � � � * � � � * ! � � � � * O �

! ! � � & � ( � � * ( � ! O � * ! O � ! & � ! O !

� � � * * O * ! � ( ( � � � � � � �

Page 212: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � # � � # � � � , # � � � � # � � � � �

� # # # � � # � � � � � � ? � � � � # � � � � � � � A � � �

A A � � � � � , � � � � � �

� � � � � ? � � # � � , � � � �

� A # � ? � � � # � � � � � ? � � � A A � � � � � , , � A # � , � , ,

# � � A � � � A � # � � # , � # � � # # ? � � , A � # �

� # # � � , A � � � , # � � � � � ? � , � � # � � � � # # #

# � � # � # � # � � A � � A � � , , # � � � � � � � ?

� � A # � � A � , , # ? � ? � � � � � # �

A , � � ? # � ? # � � A � � � � � � � � � � , , � � ?

� ? �

# # � ? ? � � � � � # ? # # # ? # � � � � ? � � ? A

� # ? � , , � ? # � ? � # � ? � � � �

� � ? � � � � # �

# � A � � � � # � , � A � � � � � � # # � A � � ? � ? �

� � # ? � �

� ? � � � � A � ? � � � � # � � � � � � # � � A �

� A � � � � A # � � � ? � , � � ? � ? � � � � ? � , � � A

� � ? � � � # � � � , � � � � ? # � � � , � � � � # � �

# � � � � , � � � # � � � # � � ? � � ? # � � ,

� A � , � � � � � � # ? � ? � A , � � �

# # � � , , # � ? � ? # � A # � � � # � � � A A � �

# , � � � � � � A � � � � �

A � � � # � , � , � � � # � � ? � � # � � A � � , � � # �

� , � � � � A � , # � ? � � � , � � A � , � � �

� , A # � � � � � � � � � # ? � , # � � ? �

A � � � � # � � � , ,

� � A # � � � � � A � A ? � �

# ? ? � A � � � � � � � , � � ? ? � � �

� � � � # �

A # A � # � � ? � � A � � � � , , � # ? A # �

� �

A # � � � # � � ? �

Page 213: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � $ � � � � ) � * � * $ ) - - * �

� � � * � - * * $ * � � $ * - * � � - � $ $ * ) � � $ * � ) �

$ � � ) � � ) $ � * - � � � - � ) � � * � - * � � � $ � � �

� � ) � * � - * - � $ � � � � ) � � � � � � ) * $ * � ) � � ) � � � )

� � l l * l � � � ) � * $ * � ) ) * - $ * ) $

� $ � � - � � l * � � ) l � ) * * � - * * $ � � l l � � � ) ) � - * � *

) � * � - � � l * � � � ) � - * - * $ * � ) � l l * l l � * * * ) $ ) l * $

* $ � l $ � $ � * � � * � $ � � * ) � * � � � � l

* * � � � � � l * )

� � � ) � � � $ � � � - � � � l

� � ) � *

� � - $ � � l l * * ) � ) � * l l � * - * � � � $

� � � - � � � l * � � ) � � ) � $ * ) � � * l l � * - * $ * � $ l * - l

* l l � - * - * � * � * � * - $ � � � * l l � * *

* $ ) � � � � ) � � ) ) � * � * ) � � ) � - � l l

� ) � * �

� $ � $ * ) � � ) � � ) * � * * ) � � ) �

* * l l * * - � � * � � � $ � l � � * � � - * - � � ) � * - � � $ � ) � � �

) * � - * � l l � ) � � � l * ) � � � � � ) * � � )

* � - * � * � ) � � � � � � $ � $ � � � - � � � l ) � � * � � $

� � � ) � ) � � l � - l ) � $ � * $ ) � � �

) � � $ * $ � � * � � � l l � � �

� - * ) � $ � l * ) *

* ) � - �

* � � $ * * * l � � * * - � $ * � - * � * ) � - � � � l * � - � ) *

� � ) � � � * � - * � * � ) * * l l � - * - * ) � � � � ) $ � l � � � �

� - � � l $ � * l * � * � ) � - � ) � � * � � ) � ) � * - * l � - * - � �

* ) * � $ * � � * � - � � � l

� � - � l l � ) � �

� �

� $ * ) � � ) � � - * � - * � $ � � � � - � � � l $ * �

Page 214: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � " # � & & " # " � � + � + & � � # " � � �

" � � # + � + & � # " + � � � # + & � � #

� � # " # � + � � � + " " � � � O � � � � O � � # + � � O O " # � & � � � �

� � � + & " " + � � & # O � � � # + " � � � & # �

" + � " + " � � & � & # � + " � # " + � � #

" � � & � " " � � + � # � � # � � # � � � � O

" � � � " � � � � � � � & #

& � � # " � � � & � " " � � &

+ � � & & � O & + � � �

" � O O

� " + + � � � � � � # � O + � � + &

� � # � + + " # + � � & + # � # & + & " # "

� # � � + � & � + & + � " � � +

� & � + � " � � + � O � � + � � " � O � " " � � � & " " � � O O

" � � � � � � � # & # # " � & O + � # & " � & # O & # & � +

� � " � � + � � � � & � " # O " � & O O � O � � " �

+ � � O � " " � & # O � + # O " � O O " &

� " "

� � O � � " " � & # � O & � O " � � & " O O " & #

& � O O � & # � O

& " � O O � # & � # � #

" � O O " � # � O � # � " �

& # # " # � " # O " + & # O O # +

� � � " � " & # & O + # � " � " " #

� & # + � � � & � # & + " # " � & # O O + " & " # "

+ � " " # � " � " & & � � � �

" � & & " � � + � # � " & + # � " �

� � # " + & � " " � & " + � O + " # " � & # O O " �

� + � � " � � " # � " � # & " � � & � " " & " # & # & & O #

" + � � " + � � � & " O " " � # � � O & " " " & " # # & + & O

� # " + +

� " � "

� & � � +

" � " �

� � � " � � � O " & O # "

Page 215: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � & ' � & ' � � � � � 4 5 � � 4 �

� � � � � ' ' � 4 � 4 � � � � 4 5 � ' � 4 � ' & � ' � & 4 � � & � 4 � �

� � ' & � & � ' � � 4 � � � 4 � 4 � '

4 5 � � 4 � � � ' 4 � ' � 4 � ' & � � & � ' � 4 � � �

4 5 � � � � � � � � � � 4 & & � � � � � 4 5 � 5 � 4 & � � � � 4 5 � � � & 4

� � � � ' 4 �

� � � � & ' � � 5 � � � � & � � � � ' & � 4 � � & & � � � � �

5 � � � � � 4 � ' � � & ' � � � � ' � � � ' � � � � � & � � 4 5 & 4 � �

& � 4 ' 4 ' � � � ' & �

� � � 4 & 4 ' � � � � ' & � � 5 � 4 5 4

� � ' �

� � � � � 4 4 5 & 4 & ' � 5 � 4 5 & 4 4 5 � � � ' � 4 5 � ' � � �

� � � � 4 5 4 � � 5 � ' & � � � & � � � ' & � & �

5 � � � � � � ' & �

� ' � � � � � � � 4 & 4 � 4 � � ' � 4 5 � 5 & 4 � 4 5 & & '

& ' � 5 � 5 & 4 4 � � ' � 4 5 � & � � 4 5 & 4 5 � ' & � � � ' � & � � � � � �

4 5 � � � � ' � � � 5 � � � 4 5 & 4 � ' 4 5 � � 4 & � 4 � 4 5 � � � � � 4 � � '

� � 4 � ' 4 5 & � 4 5 & 4 � & � � � � � 4 � � � 4 5 & 4 & ' ' 4

� & � 4 �

� 5 & � � � ' � � � 4 � � & � ' 4 5 & 4 4 5 � � � & � & � � � & � 4 5 � �

� ' � 5 � � 4 � & � � ' � � & � � 4 � ' 4 � � � � �

� � � 4 & 4 5 � � � � � & � � 4 5 � � � � � � & � � � � � � � � 4 5 � � � � ' � �

5 � 4 5 & 4 � � � � ' & �

� ' � � � � � � ' 4 � � 4 � '

5 � � � � � 4 5 4 � � � � � � � � ' � 4 � ' �

� � � � � � � � � & ' � � � ' & � � � � � 5 & �

5 � � � & ' � ' �

� � & � � � � � 4 5 & 4 4 5 � � 4 � & � � � � ' & � � � 4 & �

4 4 5 � � � ' 4 & � � 5 � � � � �

5 � � � ' 4 � & � � � & � � � � � 5 � � ' 4 �

4 & � � & ' � & � � & � � � � � � 4 5 � � � & � 4 � & � � � �

� & ' 4 � � � ' & � 5 � � � � & � & 4 � � ' � 4 5 � � 4 � � ' � �

� � 4 5 � � � 4 � � � ' � ' � � ' � ' 4 � � � 4 & 4 5 � � ' & � � � &

� � � � � � � � 5 � � � � � � � & � � 5 � � ' 4 � � � � � 5 � �

� & � � � � � � � � � � � � 4 � � � � � ' 4 � ' & ' � � ' & � � � � � � ' � � � � � � �

& � � � � � � � � � � � ' � � ' ' � � ' & � � � 4 � � � ' � '

� '

Page 216: Goldrei.classic Set Theory for Guided Independent Study

� � � # $ % # ( + � # -

< # $ � #

+ � � - - $ % $

� � < � O $ % � # < - � # � -

< < $ % � (

O + � < - O � $ # $

+ - � < � < - (

� - � O - < # � � % � - � $ % $

$ % $ � - $ � O - < � O $ $ � < $ $ O < O #

$ # � + < � - � � % O $ - $ < � � � ( $ � % � # <

$ � - � � � � O

� � % $ � - � # - $ � $ % � # < ( � # + � - � - $ � - � � ( # $ # < (

+ - � < - � O $ % � # ( - < # � - O + � # � $ % � � ( $ � �

$ - $ � O + O � + < ( ( - + $ + < O �

+ � $ - $ � � # # � - # $ % # < (

� � # � + - < - % � � % �

$ < O � + < � % $ � + < O � < # $ % � # < ( � O - � < # � # $ O O �

� � + < � O $ - + $ +

# � % $ # $ � � % ( ( < #

- O + � + � # � � < # � % $ # $ %

+ � #

$ - $ - � O < O � # $ � � - � -

% � � % �

$ % - $ - � O O � $ % � # < ( � - < # � % $ # $ % � ( ( - $

O + < - $ O � % � - � � � � % � < �

+ � # $ % - $ � - - < # � % $ # % �

� � % - � � < # - < + - < #

- $ O

$ � + � $ � � � - # $ % � # ( + � # � -

< < # $ % # < (

O - � - - $ % $

� � + � � $ # O % $ � $ � O % � < ( � # � � % � # $ $ O $ � - % � �

� � # O $ O - $ $ % � # - � # $ % $ � % � - � + O < - O + � $ % # - % �

# � < % $ % � % � O + � # � � % $ � # < � - � O + � % < # � ( � � # O $ O + � $ % � %

$ < � - $ % $ < ( O $ � # $ O + � $ % � % � $ % � # < ( � % O � # ( < � <

Page 217: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � ! � # $ � � � � ) � , # �

$ , � , J � � # ! � # # $ , ! $ , � $ , � ! �

J # � � � � $ ) � � , $ , ! � J ! # ! # � # $

� ) � � ! � , # , � J $ ! $ $ $ , $ � ! # � � ) $ � � � � �

� $ � $ � $ ) J � � � ) $ � $ � ) $ , � ! � � � � � ) J � � � ) � � � # �

� ) � � $ � � $ $ , � � ! � J $ ) ! ) # ! � ) $ � ! � � $ � �

$ � $ � � ! � # � ) � �

� � ) ) � $ , ! � $ � $ � $ � $ � � )

!

� ) � ! � $ � ! �

! $ # � J ) � # ! � $ , � � ! $ J ! )

$ # � ) J ! J $ ) J ! ) � ! � � ) $ , � � ! �

! � J � ! J � � � # $ � � � ) � ) � , � � $ � $ $ !

! # ) � � � $ � $ ! # � ) $ � � � ) � ) � , � $ � $ $ ! � � � ! ) � ,

$ � � J � � � � � � � ! ! � $ � � � $ ) J $ # $ � , � � $ � $

� J ! � � � , � � ) $ J � , ) ) $ # $ , ) #

� , ) � � # � J � � ! � � $ ! � $ � � � � � $ # J $ J !

! �

$ # � ) J ) � � $ # � � # ! � $ � � ! � �

) J � J � #

$ # � ) J ! ) J � � ! # ! � � ) $ � ! � � J ! � )

$ ) ! ) #

$ � ) # ) , ) J � ! , � � ) � � ) � � , � � $ $ $ $ � , J � $ �

) � # � � $ J ) $ # J $ ) J ! ) J � � � ! ) � , $ � $ ) J � � $ $ , � ! � #

# ) # �

, # ) # $ ) ) ) J � � ) $ $ � � # J ! # � � , $ #

! � � $ ) � � � ) J � J # ) J � # � ) J

� , $ �

$ � # ) # J $ J ! ) # ) J � , $ � � � � � ) ! � J �

, $ � $ # J $ � ) J ) � $ � � � , $ , � ! � � # ! � , $ � ! � ! �

# � � � � $ � J � $ ) J ! ) � # � � ) � � )

$ � # J # # � ) ! � $ ) � ) J ! � # ! � � , $ � $ ) J �

# � $ , � � � � � � # J � � � ! # ) � , $ $ # � $ # )

� ! � J � $ , �

� )

Page 218: Goldrei.classic Set Theory for Guided Independent Study

� ( ) * - . * * ( - 3 ) . ( )

3 ) * - * . . ( ) 3 H 3 ) H * � . 3 (

� H . ( ) 3 H

� 3 * * - � . 3 � * l * ( - 3 ) . ( ) 3 H

( - � * - 3 * 3 - . . ) 3 H - * � * . � 3 3 -

* . � ( - 3 ) * . ) . - � * * ( - - . *

. � ( � - - H * 3 * � * l * 3 -

- . � - 3 - l * ( l * . - � . � 3 ( 3 H ( (

.

3 * � * - . * 3 - . . ) - ( ) . l ( ) 3 H - * � * ( ) * * l

. . ) . � � 3 * 3 � ( . . l � ( H 3 - - . * H H . l * l * - * . ) 3 (

. ( ) � * * ) ) . � - - * ( . ) * - � . * � 3 � * . l ( ) H -

( ) * 3 ) 3 - * � * � * l * � 3 3 ) . l ( ) 3 H

. l . l � . � * . l * l - . . l . ( ) � 3 * l

( � l * * � 3 - 3 * - H * 3 ) - l � . l 3 * H H . l * l *

- * * 3 ( - � 3 *

� . - * * l * . ) 3 H * . - 3 � 3 ( 3 - 3 H H * * H . l * l * *

� � . ) - ) ( 3 H - * * ) . l * . . l � . * 3 . � * . l * - -

. l 3 H H . 3 H � * l * - H . � * . l *

( ) * * - . � * . l ) 3 H . � . � 3 � * l * 3 ) - * ) ( )

- * * . ) 3

. l * � - l * H ( H * 3 ) l * � 3 * 3 ) - * * . l *

( . - � . �

. 3 � � * ) - * H H * . * l ( . . l ( . * * 3 - * l

. . � - * - � ( � * . l * ( ( . H * � * . ( ) H

� l ) . . . l * ) . * 3 � * l * - 3 ( ) 3 ) - )

� 3 * � 3 * - 3 . � * 3 ( 3 ( . ) . � * . l * ( l * l * - ) .

. ) H � 3 3 * . ) - � ( � ( - ) . H * 3 � * *

(

Page 219: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � ! � � � � � % ! % ! ! � � � & � � � � � � � + � + ! � �

% � � � + % ! � & ! � + � � + � � � � % � � � +

� � � & � � % � � � � � %

%

� � � + � ! � + % � % � % & % � � % � % � ! � & � & � � % �

! % � � ! �

� � + � + ! � � + % � � � � + � � � & � ! � � � � % � & � � ! � � � �

� � � � % � � & � � � � % ! % � � & ! & + % & � � � � % � � �

+ � � + � � � � � � � � � � � + % & � ! � ! � & � ! � � � � � � � � % � &

� � � � � � � � � � & � � � � � � + � + � ! � % ! � � + ! � � + % � �

! � � � + � � � % � � � � � � % � � � � � � � � � � + � + % � � + % � % � �

� � � � � � � % � � % � � � ! � � & + � + � � � � � � �

� � % � � ! � � � � ! ! � & � � � � � � � + � & � ! � � � �

� � � � & � � % � � % � & % � � � � � � & � � � � & % �

% ! � � ! ! � � � � � � � � � � � % � ! & � � � � �

� � � � % � � � � � � � � ! � & � � � ! � � & � %

! � � � & & ! � � � + � % & � � � � & ! � & % � & � ! � � & � � � � � �

% � � � � � � � � � � � % � � � � � ! � % % � & � � & � �

! � � & % � & � � � � % % � � & ! � � & � & ! � & ! � � & � ! �

� + % � � � � � ! � � � � � � + � & � ! � � � � � � � � � � & � � � � + % ! �

! � � � � & � � � � + � � � � � � � � � � % � ! � � & ! � & % � � � � �

� � � � ! � � � � % � � + � � � � � � % � � + � � � � � � � � � �

� � � � � � � � � � % � � � � � % � � � � � � + � � � � % � � � � & � � � �

� & � + � � � � � � % � � & � & � � ! � + % � ! � � � & � � + �

� � � � + � �

� � & � � � � � � � � � � + � & � � � + � % & � � + % � & �

! � � � � � + � + ! � � � % � �

� � � & � � � � � � � � � � ! �

� � + � % ! ! � � � + % � & � � � �

� � & � ! � � + & ! � % � % � � � � � &

! � � � � � ! � % ! ! � � � & � � � � ! � � � & � � �

� � � � � � � � � � & �

� � � � � � � � � !

% % � � � � � � �

� � � � � � � � � � ! % � � ! � � ! ! � � � � � ! � �

� � � + � &

� + � � % � � ! � ! � � & % ! ! � � � & � � � � + �

! � � � � � � � � � � � � ! & ! � � & � + &

Page 220: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � # � # � # � � � � # � � �

� 8 # # � � � � � # � @ # � � � � � � � � � � � �

� @ # � � � 8 � � � � �

� � @ # � � � � �

� # 8 � � � � 8 � � � � � @ # � � � � � � # � � � � @ @ # � � � � � � @ 8 � � �

� � # � # � � � � � @ # � � � � � 8 � � � � � � � � @ � � � �

# � � � � � � � � � � � � � 8 � @ � � � � 8 � � � � � � 8 � �

� � � � @ � @ � @ � @ � � � @

� � �

8 � @ # � � � @ # � � � @ � � � � � 8 � � @ � � @ 8

� � 8 � � � � 8 � � @ � � � � �

� 8 � � @ � 8 @ # � � � � � � 8 � � � 8 � � # 8 @ @ # � � � � � � � � � �

8 � � � � � � � � � # � � � � � � � 8 � � � � � � � � @ @

� � � 8 @ � � � � � 8 � � 8 � � # � � @ # 8 � � � � �

� � 8 � � � � � 8 � � � � � � � � � � � � � # � � � � @ �

# � � � � � � @ � � � @ # 8 @ � � � � � � 8 � � � @ # � @ # �

� � � @ 8 @ 8 � @ 8 � � � � � � � � @

� � � � @ # � � � � � � � � � � � � � � � �

@ � � � � @ � � � � � � @ 8 � �

� � � � � � # 8 �

� �

� � � �

� �

� � � � � � � �

#

� 8 � # � � � # � � � 8 � � @ � 8 � � � # �

# � # � � @ # � @ # � � � � � @ � � @ # � 8 � � @ # � � 8 �

� � # � � � @ � � � � � � # � � � � @ # � � @ � � � � �

� � @ # � �

# � # � � � �

# � @ # 8 @ # � � @ � � � � # � � � 8 � � � �

@ # � � � � � � � � � # @ # @ �

� � � � � � � # �

Page 221: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � # � � % � � � � � � � % � � � � � � � � %

� � � � * # � � � � � � � � � * * � * * � � � � # � �

� � � � * * � � � � � � � % � � % � � � � # � � � � � � � � * % � % � #

� # � � % � � �

% � � � # � � # � � � � � * * � � � � � # � � � � � � � � � � � � � # � � �

% � � � � � � � � * � � � � � � # � � % � � � � �

� * * # # � % � � % � � # � � � � � � � * � # � � % � � � � � % � � � � �

% � � � � � # � � * * # � � % � � � �

% � � � � � # � � � # � � # � � � # � � * � � # % � * * � � � � � �

� % � � � � � � % � � � � # � % � # � � � � # �

# � * * � � � � � � � � # * � � * * � � � �

� % � � � * * # � � � � � � % � � � � � � � � � % � � � � * # � � � � � # � � � � �

� � � � � � * * � # # � � � * * � � � � � # � � # % � � � � � # � � � � # % � �

� � � � � � * # � � � � * � � � � � % � � � % � � � � � � # � � � � * * � � � � � * # % � %

# * * # � # � # � % � � % � � � � � � � � � � � � � � � � � # � � * * � �

� � * � # # � � � � * * � � � � � # � � # � � % � � � � # � � � # � � � � � *

% � � � � � � � � * � # � � � % � � � � � � � � � � � � � � � * �

� * * � � � � � � � � � � % � � � � � � � � � * * � � �

* % � % � � � � � # � �

� � � � � * � � * � � � � � # � �

� % # � # � � � % � � � � � * % � � # � # # � �

# � # �

% � � * � % # � # � � � � � � � � % � � � � � % � � % # # � � � �

# � � # � � % � � � % � � � # � � � � � * * � � � � * � � � � � � * #

� # � * � � � � � % � # � � � # � # � # � # � � � � � � � � � � � # � � � # � �

% � � * � � � � � � � � � � � % � � � � � # � � � # * # # � � � � � * �

� * � � � � # � � # � * � � � � � � * � � � * � � � � � * # � � � � � � �

� � � � � � � � � # � � � � # * # � � � � � # � � � � � # � #

� # � � � * � � � � � *

� # � � # � � � � � % � � � � # � � # � � � � * � � � � � * # � � � � � � # � � � � � #

Page 222: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � ( � � � � � � � � � �

� � ( ( � � � � ( � ( � � � � � � � � 6 � � �

� � � � �

( � � � � � � � � � � �

� � ( � � � � � � � � � � � � � � 6 � � �

� ( � � �

� � � ( �

� � � � � 6 � �

� � � � � � ( � � � � � �

� � � ( � � � � � � ( � � � � � � � � � � �

� � � � � � � � � � � � � ( ( � � � � � � � � 6 � � � � � � � � �

6 � � � � � � � � � � ( � � � � � � � ( � � � � � ( � � � � � � � 6 � �

� � � 6 � � � � ( � � 6 � � � � � � � 6 � ( � � � � � � � � � (

� � � � � � � � � 6 � � � � � 6 � � � � � � � � � � � � �

� � � � � � � � � ( � � � � � � � � � � � � � � � � �

� 6 � 6 � � � � � � ( � � � � � � � � � � ( ( ( � � � � ( � ( � � 6 �

� ( � � � ( ( � � � ( ( � � � � � � � 6 � � � � � � � �

( � � � � � � � � ( 6 � ( � 6 � � � 6

� 6 � � � � � � � � ( � � ( �

� � � � � � � � � ( � � � � 6 � � � � � ( � �

6 � � � � � � � � � � 6 � � � � � � � � � � � � � � 6 � � � � � � � � � �

� � � � 6 � 6 � 6 � ( ( � � � � � � � � � ( � � � 6 �

� ( � 6 � � � ( � � � � � � � � � ( � � � � � � � �

� � � � � � � � 6 ( � � �

� 6 � ( � � ( 6 � 6 � ( � � � � � 6 � 6 � � � ( � � �

� � �

� � � � ( � � � 6 � � � 6 � 6 � ( � � 6 � � � �

( ( � � � � 6 � �

� � 6 � � � � � ( � � � � � 6 � �

( � � � � � 6 � � � ( � �

� � ( ( � � � ( � � � � � � � � 6 � � � � � � � � �

� 6 � � � � � � � � � � � � � � � � � � ( � � � � � � � 6 �

� � � � � 6 � � � � � � � � � 6 � � � � � � � � � � � � ( � �

� ( � ( � � � � � ( ( � ( � � � � 6 � 6 � � ( � � � � (

� � � 6 � � 6 � � � � � ( � � � � � � � � � ( � � � �

Page 223: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � " � & � " ' � & � � & � � � " ' �

� � " � " � � " " � � � � � � � � �

� � � " ' � � &

� � � & � � � � � � � � � " � " � � � � � � � � & � " &

� � � � � � " � & � � � � " � � � �

� � & � � � " " � " � " � � � � � � � " �

� � � � � ' " � " � � � � " � " � � � & � " " � � � � �

� " � � � � � " � � � & � � � " � � & � �

� ' � " � � � � � & � � � � & � " ' � � &

" � � � �

� " � ' � � � � " � ' " � � � " ' � & " � � � � � � � "

' " � " ' " ' � � "

� & " � & � � ' � � " � � � � � � � � ' � � ' � " � � & � � � � ' � ' � � � �

� � � " � & � � � " � � � � " � � & � � � � " �

� � � � & � � ' & � � ' � � � � " " � " � � � & � � & & " ' � ' � � � & " '

� � � � � � " & � � � � � & � " � � & & " ' � � " � � � � � " � & &

' � � � & � ' � � � & " � � � " � � & " ' � ' � � � �

' � � � � � " � & & � � � ' � � & � � � � � � � " � � � � � � � � �

" � " ' � " � " � � " & � � � � � � � � � � " ' � " �

� ' � ' & " � � � � & " � � " � � " ' " � � � � � � � � � � � �

" ' � � � � � � � � � " � � � & � " " & � � � � �

� � � � & & " � ' � � � � & & � � � � � � � & � � � � ' " � � �

" " � � � " ' " & � � � � � " � � & � " & � � � " � � � & " �

� " � � � � " � � � � � � & & � � " � � � & � " ' � � � & " � � � �

� " " � ' " � � � & � � � � � � ' � & � " ' � � � & � � � & � � � � ' "

� ' � � " ' � � � � " � � ' " � � " ' " ' ' " & � � &

& � � � � � � � & � � �

� � " � � � " & " � � � ' " � � ' � & � � � " ' � � � '

" ' � & � � � � � � � � &

� � � " " � � & � � � � � � " ' �

� � � � " ' � � & � � � & � � " ' � &

� � � � � � � � � � � � " � � &

� � � � � " ' � � � " � � � � � � �

� � � " � ' � & � � ' � " � � � � � � � " � � � ' & " � � � �

� � � � � � � � � �

Page 224: Goldrei.classic Set Theory for Guided Independent Study

� " " ' ( 0 ( � 3 3 0 " 4 " 5 6 � 7 5 5 3 ( 4 5 ' � 7

5 ( ' " 4 " 6 " 3

0 0 � 3 3 0 " 4 " 5 6 ' ( 6 " 3 ( � 7 7 5 3 ( 4 5 � 7 6 " 5 6 � 7 5 5 3 ( ' � 7

5 ' ( 3 � 6 "

" 3 0 " ( 4 ( ( ' 7 6 ( 7 � " " " ' " " ( " 3 0 ( 5 " 3 � (

" ' 4 " ( " 4 " 6 " 6 5 " 0 " ( 4 " 4 ( ' 3 0 " ( 7 ( ' 4 " 6 6

� 0 ( 0 " " ' ( ( 4 ' � 7 " � 7 7 0 � 0 " 6 " 5 6 �

( 4

" ( 4 " 4 " � '

5 3 0 "

� 0 3 0 � 3 4 � ' ( 4 5 � 7 5 6 ' ( 3 0 " � 6 " 3 0 �

0 ( 0 � 3 ( 4 � ' ( 4 5 � 7 � '

� ' ( 7 5

' ( " � 4 ( ( 7 ( 4 ( 3 0 " " ' 5 5 ' ( 4 3 " ( 0 " 6 ( 6 "

0 � ' � ' 6 0 ( 3 0 �

0 ( 3 0 � 3 ( 4 � ' ( 4 5 ' � 7 � ' " � 3 7 ( ' " ( 3 0 " ( 7 7 ( 5 ' 0 ( 7 6

( 4 (

3 6 6 " 3 0 6 ( 0 0 ( 7 ( ( " " 4 � 5

3 5 ' ' 6 0 ( 3 � 0 " 4 " 6 7 3 ( " " 5 ( � 4 3

6 0 ( " (

� 0 ( 6 0 ( �

5 3 6 5 5 6 � ' 5 ' ' 5 3 " ( ' 3 � 7 " ( 4 ' � 7 0 " � ' " 4 5 6 " (

0 " 4 " 5 6 3 7 " ( ' " " ( " " " ' ' 6 5 ' 0

� " ( ' ( " � ' ( ' 3 � ' ( " ( ' 6 ( 3 0 � 3 5 0 5 6 " 4

5 6 ( 4 " 5 6 ( ( 4 0 3 0 6 ( 4 " 4 ( ' 5 6 ( 7 " 3 " 7 4 " 4 " 6 " ' 3 "

� " 3 ( 4 " 4 " � 5 6 ( � 7 ' " 4 " � 6 6 " ( '

" ' " 4 � " 4 " 5 7 " 6 " " � 6 0 5 " 5 ( ' 6 " � 0 5 5 ' � ( 4 " 6 ( ' '

" ( " 0 " " " � ( ( ' 4 � ( ' ( ' " ' 5 ( ' 4 (

" 3 ( 6 ( 3 ' 3 5 ( � ' � " ( ( 7 7 ( 3 0 "

7 ' " ( � 6 " 6 3 " 0 " 0 ' 3 ( 4 3 " 4 " ( � 4 3 ( 5 0 3 "

� " ( 5 " ' 3 3 " 3 0 0 6 " ( � ' 5 " 0 " 4 " 6 7 3 ( 0 5 6

Page 225: Goldrei.classic Set Theory for Guided Independent Study

' ) * , * / 2 * 4 ) * ' ; 2 ) ) * 4 ) * , * ; 2 D ' / ; ' D E / , *

; / / 4 / * * 4 ; / ' * 4 ) D 4 ; / * * 2 4 / * * / *

4 / 2 ) / 4 ) D

/ * D ) ) 2 2 * 2 4 2 * / , * * ' * , D E * 2 4 ; / ' / * 4 ; ; ' 4 * / , ; ' D E 2

/ ) ) 2 D ' / 4 2 / * E E / , * * 2 * 4 2 D , * ; 2 / / ) / , 4 ) 2 *

' ' / 2 E D * / , * 2 2 D ' , / 2 / / , ; D E 2 4 ) * * / ' / 4 ; '

* ' * , ; * / , ; ' E 2 D * / ; ' D ' * 4 ) * / 4 ) * / ; D E 4 ) * ; ,

; 2 ) * 2 *

D 2 ; ' / 4 D ' / , ' * ; ; ' 4 * E / , * , * / * * ; 2

/ , * , ; / ; / 4 * / ' D E ) 2 * 2 2 4 ) D 4 , * * * E 4 ; '

* / , / / , ; ' E 2 * ; ) 4 * ' * D ' / * , D / ' / ' / , ; ' D 2 , D 4 ) * ,

4 ) * ' D E * E E , * * 2 * 4 / , * 2 / ' ' 4 / / , * D ; 4 ; / ' / 4

2 * ' 4 ; /

/ 2 , ' / * 4 ) D 4 ; 4 ; 2 ' / * / ) 4 * , * * D 4 * 2 * 2 2 / , 2 4 / / D '

/ , ; ' E 2 ) E * ) 2 / , 2 / , D ; ' D ' ; 4 ) * / , ' E

; 4 / * 2 ' 4 / / , ) ; ) ; 4 ) * , * / 2 2 * 4 / ' D 2 ; * ' D / , * ,

2 / / ) / 4 ) * / , ' D E

) ; ' / * 4 D 4 ) * ' / ' / D 2 * 4 / / , ; ' D E

) * * D * / 4 ) , / E * 2 ) / / * * , * 2 2 4 * 4 ; ' / 2 ;

; ) D ' / * ' * * 4 / / D / E 4 * / , ; ' E 2 / , , * 2 / ' ; '

4 / 4 ) 2 E * 2 2 /

) * D 4 / 2 * 4 * 2 * , / E * ; 4 / D D 4 4 ) * * 4 ) / / * ' ; ' / ' 4 *

2 * / ' D 4 , D E * , ' D * * / ' * * ' * ) * 2 / / / ; ' D E 2

; ' 4 * , 2 4 * 2 / , 2 / E E * / , ' E 2 / , / ' * 2 ) D E E

* ' * / , D * / , ' D D ' D E E / , ; D E 2 ; 4 ' D 4 D E ' * , 2

* ; 2 4 ' ; 2 * D D ' ' / 4 ) * , / ' 2 2 * 4 4 ; '

/ * ' * ) 4 / , * ' 4 ; / ' ; * 2 ) * 2 * ' 2 E * , * 2 4 / 4 * / , D E 2 *

4 ) * ' 2 *

/ * ) * * ' * ) * ' ; 2 / , * 2 2 / D 4 D / 4 * '

Page 226: Goldrei.classic Set Theory for Guided Independent Study

� " & ) * + & + & ) 3 3 * + 3 & * & 3 3 3 ) 3 ?

+

n n n 3 � 3 n & 3 r 3 3 * 3 n & r 3 3 3 * ) n & & ? * 3

& + & " ) * + � * � & + + & �

& ? * + " & ? " ) *

r + � � 3 * ) � * ) " 3 3 ? " n " & ) & & * n * & * +

" & 3 ? & " 3 * 3 � * � 3 � & & ? " ) + " n ? " ) * +

) 3 * 3 n r 3 * " ) n 3 � 3 3 ) n r 3 � & r

� + 3 & * & 3 3 � " & ) * ? " 3

? 3 + * � r 3 ? 3 ) & � 3 n * & ? ) * 3 * ) ) & r * 3 3 3 * ) & )

" 3 * 3

� 3 ) 3 & 3 * � ) ? ) * +

& 3 � � 3 ) 3 * & 3 3 3 & ? 3 " ? 3 3 ) & 3 " ) ? 3 ?

" & & ? r " n & n 3 & 3 & * )

3 ) � ? 3 ? " 3 & 3 3 n " * n 3 " )

3 & 3 r 3 ? � 3 n " ) " ) ) & 3 + ? 3 & " ) * + ) " )

r 3 n ) & ? * + r " � + + n ? ) n " 3 * ) n & ) " r " ) ?

3 * ) 3 n & ? 3 ? 3 + " ) r n " n 3 ) 3 � + � � ) n & ) ) r 3

+ & + + & ) + � * r 3 & 3 ? * " ) * ) & * " ? * & 3 ? 3

� 3 " � � ) n ) 3 3 3 r 3 ? 3 � + & + ) n r 3 n " & n &

� 3 3 3 n

3 � r * + * n r 3 & 3 & 3 & & * 3 & ? " ) * + � � * * �

& ? + ) * ) * 3 * ) & 3 n 3 ) * ) & n

� * ) & ) * + & ? * + +

3 ) 3 & ? � 3 r 3 ? " ) + 3 & n ? * ) ) " n 3 " ) " & ) & ?

& " r 3 & ? 3 � r � ) & ? " ) + & ? * + +

3 & � & n n n r 3 3 " � * ? ) * + & * +

� n r 3 3 3 3 ) n & " � 3 ) " " & ) 3 * + & n r n

) & r r * & ? r 3 ) r 3 3 ) " & ) &

3 r r ? 3 � 3 & ? � & & ? * ? 3 ) n � r & 3 + " n r 3 r * 3

) n ?

r 3 * 3 r 3 ) r * ) * * + 3 * 3 3 ) & ) 3 * ) 3

3 n � r ) r 3 3 " ) " * � 3 � * 3 3 ) � � 3 � &

* ) r 3 * + n

Page 227: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � " � # � � � � & � � � � " � � 2 �

� # � & � � # � � # 2 & " 2 # < � 2 � & 2 & � � 2

� � � � � � � � � � & � & # & " 2 # < # 2 � � � 2 � # 2 & " < � & �

" � �

& & � � � " < " & � " 2 # 2 � � � " � # 2 � � � #

& � # < � � 2

� � 2 � � # � � < " # 2 & 2 & 2 � � # " & &

� # � � 2 � � 2 � � � � " � 2 � " & 2 # 2 � � � " # � �

& � � � � " � � � 2 " & 2 & � � & & " # < � " " � 2 & � " 2 # <

� & � " & � � � # � � " � �

2 < # � � � # � � & 2 � � � � � � 2 � & � # � � # 2 � 2 " � �

& 2

� # � 2 � & 2 & 2 � � � # 2 � � � � � � & �

2 � & " & � # 2 � � � " � � � # � � � 2 " � " & & � " � & � # � #

# < < & � � � � 2 " � " 2 & � " � � � � & � # < " 2 " � # < < � &

� � # � � � � � � � & � # � & � � � � � � < � " 2 � & 2 & 2 & �

& & # & 2 � � � � < " � � " & " 2 & 2 " & & # � & 2 � � " <

& � 2 � # 2 � " � 2 & � & & � & � � 2 " 2 & � & � � < � &

2 � � � � � � & " 2 & � � � & " 2 < " " & & � � � & 2 � � &

� � � 2 " " & & & � � � � � 2 � � & � � & � � " � � #

& � � " � � < � � � � � & � � � # � � " < � � �

� � � & � < � � � 2 & < " 2 � � # � � � � 2 < � & # 2 " �

& � � � " � & & � � " & � � " & �

� # 2 � & � " 2 # � � � 2 � � & � � � � & � � " � & & � "

� & � � � & 2 # < �

� & � � � < & � � " � & � # � � � 2 " � 2 " & 2 & 2 < 2

� & � � & 2 � � < � 2 � & � 2 � # � � � �

& # < � 2 � � � � � � � � � & < �

� 2 � & & � � & � � # � � " � � # � � � # < < � � � � �

� 2 � 2 � & 2 � � � & & � # � & � " 2 & � # < < & < < & �

Page 228: Goldrei.classic Set Theory for Guided Independent Study

& , - . 2 & , - . & & - . , ? A

O

A & - . - - . & & A ? - . , , 2 , O a ? A - , 2 a O a ? a

A . , . - . , A , O & a - . & , a & . 2 a

. O O A . & , & & A & - O . - & & - Ã 2 & & & a 2 - - . & , a

. , . & . a

. O . , a A , - . & , Ã O 2 O & ? ? - . & & & . A a 2 - . . &

& 2 & ? & 2 . & - , . , - . & A O - . & Ã & A 2 a , Ã & Ã O , , Ã O A

, , . , ? & Ã & 2 A a ? & . 2 - . O A , Ã & O A , , Ã . A

& & & a & 2 a . & & ? & & a - , - , &

, & Ã A , , . O , , 2 A

& O &

, A , A & - . 2 - , Ã ? ? . & & A 2 ? O - , ? A

A 2 O , Ã O 2 ?

A , . -

, & 2 ? & & - . & , & O A , Ã . O Ã , &

A & - Ã & Ã & 2 A ,

O . - . & A a ? A & & a - , A , - 2 - & Ã A &

& . & . O & 2 A - . 2 - 2 A & - , 2 O A O A 2 a - O , a . &

, 2 , O

. . , - . 2 & a - 2 , & O A a O A O 2 & A & - O A

. . & a O - O , a - A A - Ã . - , Ã 2 - , A . , - 2 - A , &

, , . A , - . 2 - & &

. a & O a - O , a , Ã ?

Page 229: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � # � � % � � � � - �

� - � � � � � � - � T � � � � � � � % % � #

� � � � �

T % � - � � � � � - � � T � � � # � � #

- � � � � � - � # � � � � � � � - � - � # %

� # � - � #

T - � � � % - � % � T # � % - � # � � � � T � � - � � - � �

� # � � � � # � � � - � # � � # � � � # �

� � � � � � T

� � # � � - � % � - � � # # �

- � # - � � � � - � T � % % � � � � � � # � -

� � - � � � � � # � � � # � % � # % % � � % � - � � T

- � % � � � � T � � � � - � � � � T � � � � � � # � � - - � � � #

� � � � T � # � � � � � � � � � - � T % � � # � � � T � - �

# � � % � � � � � � # � � � � #

T � � � � � � � # � � T � � # � T � % � - � � � � � � �

# � � # � � � � � # � � - �

� � � � � � # # # � � � � � � � � % � - � � T �

� - � � � � - � # # � � # � # # � # -

� � # # � # � � � � � � - � � � % � % � T # % - �

� � % � � � � � � # � � � % � - � � � � � T � � T % � - � T

� � - � T � � � - � -

� T � # � � � � # � � � - � � T � T � � % � �

� � � - � � - � � � - - � � - � � � T � � T � � # � � � � �

� � - � # �

� � � � � � � � # � � T - � � � # � � � � � % � #

� � � # � � # � # � � T � % � -

� - - - � T � � - � � � # � � � � � �

� # � � � � - - � � � T % � � - � � � � � T #

� � � # � - � # � # - � � # � # � �

T � � - � � � # � � � � � � - � � - � � � �

- � � T #

Page 230: Goldrei.classic Set Theory for Guided Independent Study

� � � � # $ � * , � � � , $ � � 4 � �

� , � # , $ � � # � 4 � $ � �

� � � � � � � � , � � � � , � 4 # � 4 $

# � 4 4 � 4 4 # $

� � # d , � , � � � � � � $ d $ $ , $ � 4 # #

$ , * $ � � �

� � � � � # d # � � � � # � # � $ � � � �

� $ , �

� � $ # , $ � �

, � � �

� � � � , � � � , � # � , � � � � $ $ �

$ $ � � � � � � $ � * � � , 4 , 4 � # d * $ 4 # � �

� , � 4 $ � , � * � � d # 4 # �

� �

4

� # d $ 4 # � , $ �

4

* � $ 4 # � , �

, 4 # $ # � � $ � * # d � * � 4 4 � 4 # 4

, � � � � # � d $ � $ # � � 4 , $ d � � � � � 4 �

# d � � � � � � # , � 4 � , � $ � � # *

� 4 4 , � � d d � $ , � $ � , � d � � 4 � , , � �

# 4 � , , � � , � 4 $ *

� # � * �

4 � * �

� 4 * � d

4 � � � � � � # � � � , d $ � � � , 4 , ,

� � � , $ 4 � , * # , , $ � � d , � 4 � � ,

� � � , � � � 4 � * � 4 � , � * � 4 $ * d

Page 231: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � & � � ) � & � * � � � � & � ) � & � * &

2 � � � ) � � � ) � & � � � � � � � � & � * & � * ) � & *

2 & 2 � 2 & ) ) � & � * ) � 2 � & � � * � � � � � � & * & ) &

� � & � � � & & � � � 2 � & 2 & � 2 � � * & � � & 2 ) 2

� � * & � ) 2 � & � � � * & � � � � � & & ) � � �

� * � � & � * � � � � � & * & 2 � 2 � � ) � � � 2 ) 2 � � &

� � 2 & � � & � * � � � & & & � & � & � � ) � ) � * � � � � * � 2

� � 2 & � 2 � & � � * & � * � � � � ) � � � � � � � �

� & 2 � � � � � � � � & � & & * & 2 ) * � � & � � � � � � 2

� � ) � � � � * � � � & * & ) � � & * � � � & ) & & * & �

& & � & * � & & * � � � � � 2 & & � � & * & 2 & � & � � � & 2

� � ) � ) � 2 � & * � � � � ) � � � � * � * � � � � 2 � � � & � � � * &

� � ) ) * 2 � 2 � � � � 2

& ) & � � 2 & ) ) � & � * ) � � � & � * � � � � �

& ) ) � & & 2 � � � & 2 & 2 � � � � 2 � 2 & 2 & � ) � � � �

& � & � � � 2 � ) � �

� & * & � ) � & � � � � � * � � � � � � ) � & � & * 2 & � & ) 2 � & �

) � & * & 2 � & � � 2 2 � � � � � � � � � ) � * & & *

� � � � �

& & � 2 & ) � ) � & � 2 2 � � � � � � � 2 �

� � � � � � � � ) � & �

� & �

) � & * � & � � � � * � � ) � � & � & ) * � � 2 � � ) & & � � � 2

� ) � � � ) � � � � � 2 � & & � � � � � � � & � � ) � & 2 � � & � )

& � & * & � 2 & & � � � � & & � & � ) � � � 2 � � � � � 2 � 2 � &

� * � � & * * � � * � 2 � & & 2 & 2 & 2 � ) � & � & ) & � 2 &

) � & � & � ) ) & * & � � � � � * & � & � &

� & � � � � � � ) � & � � � � 2 � � � � & � & � )

* � & � � * � � & 2 2

� * � � � * � �

2 � � ) � � � & & � � � ) � & � * * & 2 � 2 � ) � � ) � * ) � & � * �

� & 2 � � � & * 2 & ) � & & � 2 � ) � & & 2 �

Page 232: Goldrei.classic Set Theory for Guided Independent Study

� � � � � # � � % % & # % � ( ) � # � � � � � � 0 ( � � � 0 � # ( ) ) � (

# � � # � # � # � # % 0 � � � # & # � � ( & ( # � � 0 � ) � 0

# # ( # � # � 0 � � � ) � ) � % � � 0 ( � & ) ( 0 % � # � ) � � ( # � � � # 0 ( � � �

� � & � # ( � & # ( � � � 0 # � � � � # & ) � � 0 ( � # � � � #

� ( ) � � # # ( � ) ( ( � � # � % % � ( # # & ) � # � # 0 � � � � � # �

� � ( � 0 � � � � � # � � �

� � � 0 � � # # ) � � ) � ( # 0 � � # � & 0 # � & ) ( 0 % 0 (

� & ( � � # # ) � # # � 0 % # ( ) ( � # 0 (

� � # � � � # # ( � � � # � & #

� � � � 0 � ( � & ( % � � ( % � 0 (

# � � ( � � � � � � � � 0 & ) % � ) % ) 0 � ) � ( � � & ( 0 % �

# # 0 ( ( � & # & � � � ) � # � & # & � � ) # � � # 0 � # � ) 0 (

� ) � � # 0 ( ) % # ) � 0 � � & # % � � � � # % 0 � # & ) # %

� # # & � # ) � ) � ( � � ) ( # ( # 0 % 0 ( � ( %

# 0 ( # � & ) 0 % � � � 0 � � # � & # & & � � ) � � & # &

� � � & � � � � # � ) ( & � �

# � # � # ) ( � � � # � 0 ( � ( ) � # ) ( � ( ) ( � � # � & # � (

& # � � ) � � # � # � ) % % # # � � ) � # � � & � ) � ( �

0 & � � � # ( � ) # � � � # # ( ) � ) � ( � % � � � � # ( � ) � (

( � # # � � # � � ( # � 0 % % ( # � � ) 0 � # � # � # & � ) # � � � ) ( 0 % 0 ) � ( 0 (

% ) 0 � � ( � � % % 0 � � # � � � & ) ( �

� # 0 % ) � � ( � � # & # & � ) � ( & ) % # % # � � % � � 0 0 ) ( 0 � � � #

% 0 # � # � � � � � # & � � # ( � ) � ( # � � ) � ) ( # � ) ( � � 0 � # &

� 0 # � � # # # � 0 & 0 � ) ( # & ( # % � ) ( 0 � � � � ) � � � ) #

# ( � (

� # � � & ) ( 0 % � ) � # #

� �

� 0 ) � � ) ( %

Page 233: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � ! � � � � " � ! # � � � " � � # �

� � ! # # " � # " � � � � � ! � � � � � ! # � �

� � � � ! � � � � ! � � � ! ! � � � � � � � � � " � � � � ! � �

! # � � " � � � � � � � � � � � � � � ! � � " � � � � � � � � �

# " � � � ! � � � � � " � � � � � � ! � � ! " � � � � � � # ! �

� # � � " ! � � ! # � ! � � � � � � � � �

� � � ! # # � � � ! � ! # � � � � � � � � � �

� � � � � � ! � � " � � � � ! � � � � � � " � # � � � �

� � ! � � � # � � � " � " � � � ! � � � ! ! � � �

� ! � � � ! # � � " � � � � ! # # � ! � � ! � ! � " �

� � � �

� � � � � � # � ! � � � " � ! � � ! � � ! � � � �

! " " � � ! #

! � " � � ! # � � � � " " � � � # � ! #

� " � # � � � � � � � � ! � � � � �

# � � � � " ! � � � "

� � � � " # � � " � � ! � � � � � � � � �

" � ! � � � � � � ! � ! � " � � " � ! # " � � � � � " � � � # # � " �

� " �

� � � � � � � � � � " � � # � ! � � � " � #

� " �

� � ! � � � � � � ! # ! � � � � �

� � � � � � � � ! � � � � � � � � � �

� � ! � � � � � " � # � ! � � " � !

� " � ! � � " # # � � � " � � � � � � � ! � " � � � �

� � � ! � � � � ! ! � ! � " � ! " � � � � � � � � " � # � � � � � #

Page 234: Goldrei.classic Set Theory for Guided Independent Study

% % ) + / + 2 + 4 5 6 6 / 6 + 6 4 / + ) %

) % ) + + + 4 / 6 + + 5 % + + / % ) 5 % + 4 + +

5 % / 6 6 / \ + ) 5 + + 4 + + 5 % \ 4 / + 2

+ % ) + + + + + 5 / + 5 / \ 4 \ +

% 4 / \ 4

\ / / + / + 4 + 6 � 2 % +

\ 6 6 + 5 \ + / % ) / + 6 � 4 /

% / ) 4 6 � ) / ) ) 4 5 % + + \ / 2 6 � \ 4

+ ) 5 % % 6 5 + % ) / ) \ \ 5 4 + ) 2 / 6 % ) +

+ 2 + 2 / + + 2 6 6 ) 4 + 5 % + % +

+ \ 6 4 6

/ 6 5 % 5

+ 4 5 6 4

5 %

5 4 5 % + 4 \ 5 6 4 6 \

% 6 \ 4 5 % + 2 + )

+ + % + 5 % + 6 + + 5 + 4 / + 5 6 5 + 5 % ) 5 \ + + � 2 + + 4 5 6

\ 5 % + 4 5 % + 2 + + / 5 % ) 5 \ / + / +

/ / + 5 % ) 5 4 6 5 + + � + 2 + + 4 5 6 / \ 4 + / % ) 2 / 6 % ) % \ /

+ ) 5 6 ) 6 4 5 � ) \ 6 \ � / + 6 5 % + 5 � ) 4 / 5 \ + 2 6 / � + 6 \ +

5 % + + 5

5 % % + � 6 + 5 % ) % + 4

+ 5 + % + 6 + + 4 5 6 4 5 \ 4 % + 6 +

+ 4 ) \ 4 / + 5 6 + 4 6 4 + ) 2 5 ) 2 2 \ / + 2 2 + 4 + +

/ \ 5 % + 6 6 / + ) ) 5 \ 6

\ / 4 6 \ 4

% + + / 4 6 � \ 4 2 ) 4 ) � + 6 5 % � + � + / + 4 ) 2 + 6 � \ 5 % \ 4

6 � )

% + � + ) + + + + 4 5 / 6 4 6 5 \ 4 + % ) +

% + \ 6 6 4 5 6 6 4 5 ) \ 4 \ 4 \ ) + 2 + + 4 + 4 \ 4 5 % ) 5

5 % ) 2 6 4 \ 6 4 / \ + ) 5 6 4 6 % \ /

+ 5 % % / 6 4 5 6 + 4 + � ) 5 + % + 6 4 5 � ) 5 6 4 6 2 6

) 4 / 5 � ) 4 / \ 5 + + % + 5 % ) 5 5 % +

/ 6 5 % 5 / + % / % ) +

Page 235: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � * � � , � � , � * 2 � 2

� � � : � * , �

* � � � � � 2 � � , � � � � � � � � � � � , � �

* * � * � � , � * � � � � , 2 * � S � 2 � � � * �

� * � � :

� � � � � � * � * � : � � � � � * , � � , � � � �

� * � � � 2 � � : � , � � : 2 � � � � * � � � � � � � , � � � � S �

* � � � : � � , * � � � � * � � � 2 � �

� * , � � � � � * , : � � � * � � 2 � S � � �

* 2 , : 2 * � * � : � � � : � : � � 2 S � 2 * �

� � � * � � , � � � S � � � � , � � � 2 � � , �

* � � � * � � * � 2 � � : � S , � � � � � � � � � � � , : � � � � * ,

� : � � , S � * � � � � �

� � : � � � * � � : * * � � , * � : � �

� � * � � � � � � � � * * * � S � * � � � � � � S S

� , � �

* � : � � * � S � � � , S 2 � � 2 * � � : � � � � � �

� , S * � � * � : � � , � � � 2 � 2 � * � , � *

* , � � :

: � 2 � : � , � � � � � * � � � 2 � � � � � * 2 * �

* � � * � : � � � * 2 � : � � � � * , S S � , *

2 � * � � 2 � � : � � * � � S � � * �

2 , � � �

* � � , � 2

2 2 � : , S � � � � � * � : �

� 2 � S 2 � * : � � � � 2 � � � � * , 2 � � � , 2 � 2 �

* � * � � � S � 2 � * , S � * � �

* � �

� � , � , 2 � � , 2 � � � �

� * , S � � * � S � , � � � � � �

� � � * S * � � * � � * � 2 � :

, S S � � � � � � , � * � � � � ,

, � � : � � � 2 � 2 � �

Page 236: Goldrei.classic Set Theory for Guided Independent Study

� � � $ & ' * + � *

� � � ' & ^ � ' ' $ � & '

� � r ^ * � * � & $ $ � � * � ^ *

& ^ * ' � * � * * � * $ + � $ & ^ r r + * ^ *

� ' � + � r r � + & $ $ ^ � � � $ � & ' $ + � $

& ^ � r r � ' * ^ * � � $ * � � � � * � ' � $ +

� � � � ' $ � & '

& ^ � � � $ � & '

� � � ' $ � & ' � '

^ * � ^ * r * � ' & & r r & � � � ' $ � ' � ' � $ * � ' $ � & '

� * � $ * & ' * * ' * & $ � � r * � � � $ $ � $ * * � ^ * ' $ � & '

* � � & ' $ � * ' � * $ & � + ' $ � & ' � & r & $ + *

' $ & ' � � & & r � * * & ' � * & ' * � ' � ' $ + � $ $ + * � ^ * $ + * � *

$ & � � ^ r � & ^ � & * & ^ � ' � r $ + * ^ * � � ' $ � & ' $ * ' � �

' & ^ + � � ' � ' $ + � $ * r * � � r � & r � � $ � ' � + & ' $ + � $

$ + * ' $ � ' * � $ �

* � + � r r � * � ' ' $ * � ' � & ' $ & � + & $ + � $ ' & ' + � + * � + � r r

^ * * $ & ^ � r r � * + � * � � � � & ^ r * � � � ' * * �

$ * + � r r * ' � * ' � r r * & ^ ^ * � ^ * ' $ & '

& ^ * $ * $ + * ' $ � ' $ & * + � � � � & � r � ' $ � & '

* $ * � � * $ + � $ * � � � � ' � $ + � $ * � $ � * ' * � � * $ & � � ^ �

� � � $ ^ � � + $ & ^ � ^ $ & � + & $ + � $ � � � ' $ � & ' � ' * r * * $ � � $ & &

* ^ � � * * & * $ + � � * $ & * $ �

& * + * � � $ � & ^ � r r + * * � � � r � � & � ' � r *

+ r * $ + * & & ' � $ ^ $ � ' $ & � * ^ * + � & ^ r * �

� & * $ + * ' � & $ + � $ + $ * � & ^

* � ^ ^ * � � + � '

� ' $ + ' & & $ � � + � � + & r � * & & & � * $ � & � � ' ' $ � & '

� + & � ' $ + $ + * * $ � ^ & * $ ' * * * $ & * � ' $ � & '

� ^ � $ & � r r * � $ * + $ * � ^ * * � r ' � � * $ � � $ + � ' �

* + � * � r � * � ' $ � & ' & ' $ * � & & ^ * r � * * ' $ � ^ � ' $ * * � $ + $ � $ � � �

' $ � � $ $ � $ � � * � � * $ * ' $ � * ' * � * $

� + �

� � $ & ' * r * * ' $ & � � � ' * r * * ' $ & � & * � & � � � ' & ^ * ^ *

� ^ & � * $ + $ � � ^ * & $ + � ' + * ' $ + * ^ * � $ *

+ � $ ' � $ r & � & * ' * ^ r � $

Page 237: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � $ � � � � � ) + ) + ) + . � $ � �

� � � $ � � $ � ) $ � � � � $ � � $

� � � + $ � � � + $ � � $ � � $ � $ � $ � $ � �

� $ � � + $ � $ � � � � $ +

� �

� + � � $ � � $

� � . � � ) $ .

� � � � � � $ ) � � + � $ � $ � . . � � � + � $ � � � � �

� � � � � . . $ + � . � � + �

� � . � � $ � . � � + + $ � � � � � + � . � � � � + � � + � � �

� � $ � + ) + � � ) � � $ � � + � + � � � � + ) +

� + � ) � + � + � � � � + � � +

� � $ � $ � � . + � + � � + ) � � � � � � � $ � $

� � . � ) $ � $ + $ � � $ � � � � + + � � $ � � $ � ) � . .

� � � + ) + � $ � � $ � � + � $ � � + � � + + ) +

� � � � � $ � � $ � + � + ) + � �

$ � �

+ � � � � � � � $ � � � � � � � � � $ � � $ � �

� � . .

� � $ � � � ) + + ) � + $ � + � � � + � � $ � �

� $ � $ � ) � � ) � � � � $ � � + � + . � + $ � � + � � $ � + $ +

� + � + � $ � $ � � � � � � � � � $ � �

� + $ � � � � � � � � � � + � $ � �

� � ) + ) +

� ) + � � � � � � . . � � + + � � + $ � +

+ $ � � ) + + � + . $ $ � � � + + ) + ) � + �

� � � � � � � $ � � $ � � $ . + � $ � � � +

� � + + $ � � $ � � �

� � � � + � + � + $ + + � � + � � �

� � � � $ � � $

� + � + + + � + $ + ) $ � + . � � + + ) ) � � ) � . . � ) $ � . � � $

� � + ) � $ $ � � $

� + � � $ � + $ � � $ � � � � � � � + � . . � $

� � � � � + � � � $ � � + + � + $ � )

+ . + $ . � � + $

� � � � � � ) + � ) . � � � � +

Page 238: Goldrei.classic Set Theory for Guided Independent Study

! " % & ) ! % & % " ! & 5 5 % 7 % " 5 8 & ! 8 & 7 ) ) 7 " ?

5

7 % 5 ? ! & 8 " 8 Y & ?

7 % &

5 ! & 8 ? 8 & ? 7 ? 8 7 ! 7 8 7 % 5 ) ) 7 % ? 5 ) ! " 8

Y & % & ! ) 8 7 & % " & " Y 8 Y & )

& & 8 Y 5 8 8 Y & ! & 8 7 7 " 5 ) ! " ? & ! " &

5 Y 7 Y 5 8 " ? " 8 & % 5 ) )

Y 7 Y 5 8 " 8 Y 7 8 ! " " ? 8 " 8 " 7 ! ! " & 8 7 & ? & 5 ?

& ) " & ) 7 % & % ? & Y " ? " 7 ?

Y 7 Y 8

5 & 8 & 5 ? ! & 8 Y 7 Y 5

& 8 & ! & 8 ! & 8 7 8 5

5 ? 8 7 Y & % & % & 5 " %

& 8 Y & & 8 ! 7 ? & % ! ? 5 ! 7 ? ! 8 % 8 & " ? & % " 8 Y

7 ? 8 % 8 & " & & " ? Y & % & ! ) 8 7 & " & 7 & 8 " ?

" ) ) & ! & ) 7 Y ! 5 % 8 7 8 Y & & & % " ! &

" ! & 8 Y & 5 8 8 Y 5 7 ! Y 7 Y 5 " ? 8

! & Y & % & ! ) 5 7 ? 8 Y & % & ! ) 8 7 5 " ? & % ! &

" " ! & Y & % & ! ) 8 5 8 & & 8 Y 5

" " Y 7 Y 8 " ? ! & & % " ? 8 & & " ? 5 ! % 5 ?

5 ! " " ! 5 " ? ? " 8 & ! ! & 7 7 ! & & % & ! ) & 5 7 8

? 8 & % & ! ) 8 7 5 % 8 " ? & % ! &

" " ? 5 ? 7 8 Y 5

8 Y & % & " 7 ! & 7 ? & & ? & 8 Y & 7 & % 5 7 ? ! 7 7 % " ? 5 ) " 8 " 7 ? ?

" ) " 7 ? 5 ? & ! 7 & Y 7 8 5 ! 7 Y & ! & 8 Y & 7 % & " 5 Y " ? & % ? & & &

8 7 ! Y 7 5 8 8 Y & ! & & & & ) ) & ? & " Y " ? & ! ) ) ? 7 " & ! 8 " &

7 & 7 8 Y & % 7 & % 8 " & ! 8 & & 7 & 5 8 " 7 ? ! 5 ) 7 ? 5 ? & 8 % 5 & % 5 8 7 ? 7

& 7 ? & ? " 8 " 7 ? & 8 5 " ? 8 Y & 7 & % 7 ? & 7 % " ? ) 5 7 & %

% ! 8 ) & 8 ! & & 8 & & 7 % 8 " 7 ? ! ) 7 & ) % & 5 & & ? 8 &

& ? " 8 " 7 ? 7 " 8 " 7 ? 5 ? 8 " ) 5 8 " 7 ? " 8 & " 7 ! & " 7 ?

& ) ) " & 8 Y & 5 " ? " ? Y & 7 % & ! 7 5 % ? 7 5 8 " 7 ? 7 ? "

& Y ) ) ! & % ? 7 Y & % & ! 8 7 8 Y & 7 & & 5 ) ! 7 ! & 7 7 % &

& 7 & Y 5 8 7 & 8 " ) ) 5 & ) & 5 ! & ? 7 ? & % &

Page 239: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � � � � ( � � � � (

� � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � (

� � � (

� � � � � �

c � � � � � � � � � � ( � � � � � � � � � � � � c � �

� � � � � � � � � � � � � � ( � � � � � � � � ( � � � � � � �

� � � � � ( � � � � � � � � � c � ( � � � � � � � � � � � � � � � � � � � �

� � � � �

� � � � � � � �

� ( � � � ( c � � � � � � � � � � � � � � � � � � � � �

c � � � ( ( � � � � � � � � ( � � c � � � � � � � � ( c �

c � � � � � � � � � ( � � � c � � � ( � � � � � � � � � �

� � ( � � c � � � � � c � ( � � � � � � � � � � c � � � � � � � � � � �

� � � � � � � � � � � � � � � � c � � � � � � � � � � � ( � c � ( � � c � ( � �

c � ( � � � � � � � � ( � � � � � � � � c � � � ( � � � � � � � � c � � � � �

( � � � � � � � � ( � � � � � � � � � � � � � � c � � ( c � � � � � c ( � � � � � � c �

� c � � � ( c � � � � � � � � � � � � � � � c � � � � ( ( ( � � �

� � � � � � � ( � � � � � � � � ( � � � � � � c � � � � � � � � �

� � c � � � � � � ( � � � � � c � c � � � �

� � � c � ( � � ( � � � � � ( � � � � � � � � � � �

� � � � � � � � � � � � � � � � ( � � � � � � � � � � � ( � � c � ( �

� � � � � � � � � � � � � � � ( � ( c � � � � � � ( � c � �

� � � � � (

c � c � � � � � � � � � ( � � � � ( � ( � � � c � � � ( � �

� ( � � ( ( � �

c � � � � � ( � ( � � � � ( � � � � � � �

Page 240: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � ! � $ & � � � � & � � � $ � � � � � � � � / 0 � � / �

� � � � / � � � / � � � � / � � ! � $ � � � � / $ � & � � � � � !

0 � � � ! � � � � � ! $ & �

� $ � � ! � � $ & � $ � � ! � 0 / 0 � $ � / � ! $

/ � � � / 0 & � � � / / 0 � � / � 0 � / � ! ! 0 / $ & � � �

0 0 � � 0 � � � � ! $ � ! � � � � / 0 � � $ � � 0 � � � � $ / / � $ /

� � � � � / 0 � � / � � � � $ 0 � � � � 0 / � 0 � / � � 0 / � / � / 0 � � ! � � 0 � /

� � � � � / 0 � � � / � 0 / � 0 � � ! � ! / & � � � � � � � �

� � � ! & � 0 / & �

� � � $ & � � � / � � ! � & � 0 � / 0 / � � � ! � 0 / � ! & � & � � � ! � � � � �

� � � 0 � � / � � 0 / � � ! � � � ! � � � � � � / � !

� $ & � $ � � � � ! � ! & 0 � � / $ & / � � 0 � � � $ � � � ! � � � � & � � ! $ & &

� & � � ! 0 / $ & � � � 0 / � 0 � � � � � � � � � & ! � � � & � �

0 / $ � 0 & & $ � 0 / &

� !

! � 0 �

� � � � � � � $ � � ! � � & � � & � � � $ / 0 / & 0 / $ / � � � 0 �

& �

$ � � � ! � � & � � � & � � !

� 0 / � � � � / 0 � 0 � / �

$ / ! � 0

& � � � � � � $ � � � ! � � & � � � & � ! $ & & � � ! � 0 � $ / 0 / � & � 0 / &

� � � � $ � 0 � & � � !

� 0 / � � / 0 0 / � & 0 & 0 � 0 � /

� � 0 / � � � � � � � 0

� 0 � & 0 0 � � ! 0 /

$ � � 0 ! �

� $ / � � / � / � � ! � � / $ � � ! / $ � 0 � � � � � � � $ � � � � �

& � 0 � / � � � ! � $ � $ / � ! 0 / $ & � � 0 � ! � 0 � � � 0 � � �

Page 241: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � " # �

" � � � � � � � � : � � � � � � � � � � � " � � � � � # : � �

# � � � � � � � � � � " � � � � � � � : � � � � " � � � � # � �

� � : � � � � � � � � : � � # � � � � : � � # � � � � � � � �

� � � � � �

� � � # � � � � � � " � � � � � � # � � � : : � � � : : � � � � # � # � �

� : : " � : � � � � � # : � � � � � � � � # � � � � � � �

� : : � � � � � � � � : � � � � � � � � � � � " � � � � � � � � � : �

� � # � � � � � � � � � � � � � � � # � � � � " � � : : � � � � � � � � �

" � " # � � # : : � � � : � � " � � � � � � � � � � # : � � � � � �

� � " � � � " � " � � � � � � # � � � � � � � � � � " � " � � � � : : " � � �

� � � � � � � # :

� � � � � � � � � � � : � : � � �

� � : � � � � � � " # � � " � � � � � � �

# � # � � � : � � : � " � � � � � � : : � # : � � � � � � � � � � � � �

� � � � � " � � � � �

� � � � � �

� " � � � : � " � : � � # � � � : � �

� " � � � � : � �

� � � � � �

� � : � � : : � : � � � � # :

� � � � � �

� : � � � � � :

� � � �

Page 242: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � - 0 - 6 � � 9 9

0 0 9 0 � � 6 6

0 � -

0 0 9 � � � � � � � 0 � � � X � � X � 0 � � � � 9 Z � Z 0 9 0 �

� � - 9 � Z 9 � 0 9 0 Z � � � X - 6 � � � Z 6 � - � � 0 � � 0 6 6

- � � � � � � � � � X � 9 0 X � - � � � � � Z � � � Z 0 Z 9 � Z � � �

- � 9 � 0 - 9 � � � � �

0 Z - 9 � � � 9 0 9 0 9 � � Z � 0 � � - � � � 6 �

� � 9 � � � 0 � � 0 - X 6 � 6 � � Z � 0 � 9 0 � 0

� - 0 Z X 9 � � Z 0 9 9 Z 0 0 � - Z 9 � 0 � � - � 9 �

� � 0 0 Z � � � 6 � 6 9 � 0 0 X 9 � � - 9 0 � � � � � 0 6

0 � - � � 0 � � � 0 X 0 Z � 0 X 6 0 � X 0 � - X

-

� 6 � � 0 � � Z 0 � � � 0 6 � 9 0 � � � - � Z � � � Z 0 � � - Z � Z 0 9

Z 6 - 6 0 Z Z � � � � 6 � 0 � � 6 0 0 � � Z 0 � � - 0 Z 9 - X - Z � 0 9 � -

� 0 � - 6 6 0 - � � � 6 � � � � � 0 � � � � 0 X X 0 Z 9 � � 0 � �

0 - X 0 � 9 - -

� 0 0 � � � 0 � � 0 � Z 9 - X � � � 0 6 Z � - 9 9 � � Z 9 0

0 0 � � 9

0 � � � �

� 9 Z 0 0

Page 243: Goldrei.classic Set Theory for Guided Independent Study

� �

! ! % & ! ' % � ' ! & ! ' - ' % . & ! / . / -

/ % ' & ! ' - ' % . & ! / . % & / � � 7 � 7 � ' �

/ & ! - ' % . & -

' 7 & / � /

' � 7 7 / � /

/ & ! � & ! ' ' % & ! / . % /

' ! & ! ' - / & ! & ! ' . 7 7 & / - 7 � � . � ' / ' & ! � & ! ' - ' % . & ! / . % / -

� . . / - 7 � . ! ' - ' 7 � 7 7 & / - 7 � � . ' �

' � 7 & 7 / � /

& ' 7 � & 7 ' / & ! ' % 7 %

7 ! . / / - / 7 % � & ! / ' % ! / ! � & & ! 7 % . � % & ' - ' % % 7 / � ' %

! 7 ! ' � . %

' � ' ' % / ' ' - � � ! 7 � ' - ! . . / ' - / ' - 7 7 � % / ' ' -

% ' 7 � ' . 7 & 7 ' 7 � / . 7 � ' & ! - ' ' & � � � / - 7 � � . � 7 & ! ' & 7 � & 7 / �

! ' % ' � % 7 � . . ! & ! � & - ' . / � � % & - 7 & . 7 � - ' � '

7 & ! � - ' � � / - � ' � . 7 ' 7 & % ' ' % / - ' � � - � . & / % ' & ! ' � / & 7 / �

- � ! ' - & ! � / - & ! ' / - ' - ' . � & 7 / � / � 7 � . % & ! ' � ' � 7 � / -

- 7 & � - ' � & % - ' ' ' - & � & � � ' � � & ! ' % � ' & ! 7 � / - - 7 � � . %

' & ' / - 7 � � . % ! ' �

7 7 ! ' �

� � & ! ' �

& !� ' �

' % ! � . . - / ' 7 7 � � . ' � ' ! ' / ! ' - � - & % / - / � % � � ' ' - 7 ' -

' ! / - � � % � & ' � & 7 / � / � / - ' �

/ % ' & ! � & ' % ! & / ! / & ! � &

7 ' �

� 7 / � / � / - '

Page 244: Goldrei.classic Set Theory for Guided Independent Study

� � � � % & � ( * % & & % & & 4 * � % &

% 4 & % & % % & & � � 4 � ( & � �

% & � � 4 � & � 4 * % & ( � � * & & � * & � �

% % & & � ( * & & ( % 4 * * 4 &

% % & & & % & � ( % & % 4 &

& * � � � * � � � 4 � � *

% & & � � � �

& * � * �

� % 4 % & & % � � � �

� � & % % & ( & � % � 4 * * � * & ( & & � % 4 � �

� % % 4 � % � � � & % 4 &

& ( & � � � � 4 �

% & ( & � � % � � � �

( & ( &

� � & % % & & � � % � � � 4 � � & ( & 4 � � ( � * 4 �

� % & � % 4 � % � & ( & � � � % � � � � & * � *

� � � � * 4 � 4 * � & % 4 &

� % 4

& % 4 & 4 � ( & 4 � % � * % 4 � &

%

% � % & � * �

& ( & � � � � � � � � & ( * & � ( 4 * � * & * � �

� � & % & 4 � & 4 ( & * ( & � � & � � * 4 ( � � & & � ( & � ( � � *

& & � & * % 4 % & & * � * � � � � � � * & & * � � 4 � � *

( � & 4 * � � & � ( & * & � ( � ( 4 � �

� % & *

* & *

Page 245: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � $ � $ % � � � � , � % % � � - � $ � $ � - � � $ - � � $ � $ �

, � � - � � � % - � � > � - , � > � � $ - � � � - $ � � �

, � $ , � $

� � � > %

� � - , � , � � � % - , � % � � >

� � � , � - - , � > � � % , � % � - � � $ � $ � $ , � � , � $

� � $ - � > � % � >

� - , � , � � � % - � % � � >

� $ � % � � � � - , � - , � > � � % - , � % � � � % % , � � � � � % � � - � > � $ � %

� - , , � $

� , � $ � � �

� � � � � % � � - � $ � � - � -

� $ � - � � �

, � , - > � $ - � � � $ � - , � > � � % �

, � � � % - - , � , � � � % % - > � $ - � � $ � � $

, � � > � - � % � � > > � � % - � � � � � � � � � � - , � � > � > - > - > �

� , � � $ � � � $ � - � $ �

� � > � $ � � $ � � � � - � � $ � � � � � � � > � $ > � , � - � � > � > �

� $ � - � � � - - � > � $ % � � - � � $ $ % � % � � � � � � $ � � � - � �

� > $ � - � � � $ , � - � -

, � > � , � � > �

, � - , � -

Page 246: Goldrei.classic Set Theory for Guided Independent Study

" ' ( '

; = " ( " ' @ A ( B C ; B @ " A C = " ( B ( " @ ; A N A ( " ( '

= B ' @ C ; ' = B N

" @ " B " " " @ ; @ A = N ' B @ ' ; ' A =

( " B A = " ; N B B @ " N = ' ' ; = ' A ( " N ; B ( @ " " B C ( " C

' N = " = B " = " B " " " @ ; ( ' = " @ A ; " A N A ( = "

' @ " N = " @ " N @ ' " @ A " ; ' ' @ A ( C ; " ' C ' ( A C C ; N N A

= ; B N N A " " N =

= " ' @ " ; N = " ' @ ( B C A A " C B " @

' ( ' N " ; " ' " C ; ' B A A ; C ; " @

' ( @ ; N " ; ; ' N = B A B ( " @ ' ( = ' C C ' ; " B ; A C ;

' " B = ( " ; B C ; ' ; = ' = B N ( ' B C " @ A ( B C A

B @ = ; A ; B C ; ' " B ; ; ' @ B B C C " @ @ A ( B C " "

' = B A ; " ' ( ' @ N = ; " N = A = " " B N " @ ' (

' = " " N B @ " @ "

B = ' N = N

= ' = B N ' @ A ( B C A C A A A ' N A

A " " C " ' ( ( " @ ' ' @ ( B C ; ' @ A

; A ' @ B ( " A N = ' B ' ( ; B ( ' @ ' @

" N " ' @ A ( B C ; ' = N A N " ( " ' N " ' ' A ' C

B

" ; ' C C " C " A ( " A " @ " " " ' @ '

' @ A A N ( ; A B N A " " = B ' ( N B

N = B ' N ; = ' ( N A ; B C C "

" B C ; A @ " @ A A ' ' A '

' @ " " ( " @ C ( C ' " A ' ( ; N B ( N ' ( ( A '

' N " = A C " ( N A ' (

Page 247: Goldrei.classic Set Theory for Guided Independent Study

� � � ! " $ % ' ( % % � $ ! % � � � 4 ' $ �

! " % 4 ' N � % $ 4 $ � ! � 4 � $

$ ( ! % � $ % ! ( � 4 ' ' $ $ � ! $ % 4 ' $ % � ( ! 4 $ % � � % ( ( 4 $ � !

� � $ ! %

N ( � N ! � $ ! % ' $ ' $ 4 � ! % % � ! % � % � � $ ! % $ ( ! % � $ ! $ � �

� % ( � N � � � % ! % (

! 4 � 4 " � ' ' ! % � $ % ! ( 4 ( � ! � � 4 � � 4 � � % � ! ' $ $ �

! " % 4 (

� � � ! " $ % 4 ' 4 % 4 % $ ! % N � % $ ( ( 4 $ � ! � 4

% ! % $ ( � " $ � ' % � 4 ( $ % 4 % ! % $ % ! (

N � ! " � � 4 N ! � $ ! % ' � $ ' $ � $ ! % 4 % � ! % � % � $ 4 � $ ! % $ ( % "

4 ' $ N � ( � % ( � � N 4 � � N � % � ! % (

! !

' ' '! 4 � 4 � % ! " 4

� � ! $ % ' ( 4 % % ! 4 ' % � $ ! N � % ! " 4 %

% ! % � ( � ! $ % ' (

N � ! " � ( % ! " $ % ' 4 % $ ( ' � N � ' � � � ! % !

N � " ! ! ! % � " ( � 4 " 4 � � ' � ! ! $ $ ' $ � � ( ! " � N 4 � $ � $ ( $ 4 %

N 4 � $ $ ( % �

! % ( � " � N � � N � % N � % ! %

$ (

( ! � N � 4 ( ( $ " � $ % � 4 ( � $ � ( % ! " 4 '

( � N ( � 4 % ( � � $ � N � ' � � � " ! % !

( ! � N 4 N � ! �

Page 248: Goldrei.classic Set Theory for Guided Independent Study

� �

, - , -

6 7 � 8 - � - ; � - � - � - 7 B � - , - � C 6 - , � - 8

8 7 ; C 6 7 ; 6 7 8 7 � � 8 B 8 � 7 , ; - B - 7 ; -

6 6 � - - - ; - 8 7 B - , - � B - 7 � - � -

6 B � ;

C

C

� � ; 7 - , 6 7 6 � C

8 - 7 ; 7 8 B �

C

7 B - ; - , 6 7 6 6 B ; B B - - 8 � 6 - B �

8 � 7 ; 7 - , 6 7 6 � 6 B � ; - , - 8 � � 6 -

8 B � � 8 � 8 7 , - ; � 8 7 � 6 B

� 6 � 8 � 7 6 B ; 7 - , 6 6 , � 8 � � - C - �

- , 6 7 6 � � - B � B � ;

, - -

- - 7 7 ; 6 � 8 7 6 6 C - - B � - , 6 6 ; B 6 , 7 ; C ; 6 7 8 ; � �

; 8 - ; 6 6 - ; ; , - 6 6 - , 8 - 7 B - 6 6 , 8 7 ; C ; , 8 B � - 8

6 � - B � ; ; 6 , 8 7 C � ; 7 B B 8 � C 8 , 8 7 C � B � ;

- � ; C - C 6 8 � - 6 , - ; - B 6 - B - 7 B 8 6 - 7 -

6 6 , 7 ; 7 � 6 - ; , 8 B , 8 � 6 � - 6 8 7 C � - ; - C , - ;

6 - - ; 6 , C 7 B 6 7 7 C - B - B � - -

Page 249: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � �

� � � 0 � � 4 5 � � 0 � 5 4 � : 0 � �

: � 0 � � � � : � 4 5 � 4

� � � � 5 0 5 � � � � t : u 5 5 0 � t 5 � u � u 4

� 0 � � � : 0 5 � � � � �

� 5 t �

t � 5 0 � 0 � � � � t u 0 � � � 0 � t � � 4 0 : � t : t 5 � � 5 � � t : 0 5 � � 4 : �

� 0 � � 4 u � t � � 0 t : � 4 � 5 0 � 5 � 0 5 � � � : � � � � � � t � t : 0 � t

: 5 5 : t u � � 5 � 5 � � 0 5 0 5 � 4 � t � t 5 � : u : t 5 0 � � 5

� u � t 5 � : u 4 �

5

5

5

� 4 � : u u � � 5 0 � � 5 4 0 t 0 5 � u : : � u � : � � � � 0 : 4 � � t 5 4 � 4 �

4 � : u u 4 � � � 0 � � 0 5 � � � � � � : �

� � � �

: �

4 � � : 0

5 � 0 � � � 4 0 � �

Page 250: Goldrei.classic Set Theory for Guided Independent Study

� � � � � ! � � � " � $ � � � � ! * + * $ + ! * � � � ! �

� �

� * � � �

� * � � � + ! *

* � � " � � � �

� ! � � + ! +

� + + � * �

" � ! � � � �

� � � � ! � � " � � $ � � " ! $ $ � " � + � ! $ � � " * !

� � ! � � � $ � � "

� * + � * � ! $ + +

� � * � " ! $ � $ $ ! � + *

� + * � � " � � � " ! $

� * � " ! $ + �

� * � � � $ ! + �

! � � " � � � �

� � � � " * � � * � + � *

" � � � � � � � " � � � " ! $ $ � " + * $ ! *

!

+ * � " � $ � � � � ! � � � � $ $ � * � ! � � � * � �

! $ " + * ! $ �

� " ! $ $ � " + ! $ �

� � � � � ! * � * " � � ! ! *

� � * + � � � + + � $ ! � " � � � � + � � �

� + $ � � " * � ! * � � � + �

� �

� � " � $

* � � ! ! + � ! $ $ � � " * ! $ � �

� $ � � � � + � ! � � " + !

� � � ! � � � + * � ! � � � � * � � * � * � � "

Page 251: Goldrei.classic Set Theory for Guided Independent Study

� " $ % & / % & & " % & � % = % @ $ % & $

A $ @ $ A @ $ % " @ " % & � %

" @ $ % & % / % & & " % & � %

& " % & � %

� � = % & $ & = % $ & � $

$ / $

$ A / @ $

% & $ % % / $ ± �

& / @ " $ @ � @ A $ / A $

$ & � $ � ± $ � @ & " = & $ A $ @ ± = % $ � @ $ % %

@ " % & � % = % & $ $ @ ± % " � A % � � = $ @ $ $ & � $

& / @ $ $ A / @ $ & " % & % / = $ % � / = / A @ % $ @ / = = " / � ± @ ± ± $ A

% & � = % " % & $ " " " A @ / ± / $ @ $ $ & @ " $ @ = %

� ± @ & " ± A % @ $ @ % @ % � " ± / � % $ " / � ± @ @ � = $

@ ± $ = " @ @ " = / % = " $ @ & & $ $ $ " = $ " $ @ A " ± $ %

" & % & @ / @ / = $ $ % & $ $ $ @ & ± ± @ $ $ ± $ A / = $ $ / $

/ $ $ $ " = = / % $ " A / = � ± @ / % & " / &

& $ % & $ % / " " A / = ± @ & � @ $ � $ ± $ = $ % " = $ / & % $ ± ± @ & � %

" % & $ ± $ & � @ % @ " A % & " � / = " = � ± � A / % & $ % / & % � % "

= $ � � % & $ � % / @ & $ ± � % $ A = " ± A � % ± � ± / � % " @

@ " A " / $ @ / " % % / = @ / & @ / % " % & $ @ % % A $ " � = % " A % $

% " " % % & $ " A = @ � � " " = / = $ " = % & $ = $ $ & $ =

% $ % / = � A " $ @ " $ % & � = = $ ± � = % $ @ � = = $ $ $ / = @ / & % @ / = % "

% & $ % A A $ " = / " A / = � ± @ " A / = % � = $ % & $ ± @ � " = " = " = $ "

" � $ / & $ @ $ @ " A @ " A ± / / @ $ @ " $ = @ / & % " & / @ @ % A $

= $ $ $ " " A / % & $ / / @ & � % % / $ @ A & $ / = @ & @ = " % & $

Page 252: Goldrei.classic Set Theory for Guided Independent Study

� � � " # $ " � & " � # � � � & & � / � & � $ � � �

# � $ & " / #

$ # / " � / # � $ " & / B " " & / # B

$ � " $ # & � � � # � � $ " " & / " # B " $ � & � � # $ " " $

# # & � �

/ � & � & � � # & � � h � $ B " & / h " # B # �

� / h � & � � � # �

� $ � B B � " � � # � # � h $ & / $ � � &

� � h � B & / " # B $ # " / B / $ / � # � B " � #

$ / $ � � & # � $ B B � / h $ & � & � � & / B " " & / # B $ &

� �

/ � � & & / � � / # " " / # $ " $ B " / h

# � � $ / & $ h B # $ / #

/ # & " & � $ h / $ $ � / B � � & � &

/ " & # & � / $ � � � " # & � � � � $ " / # / �

/ h & � $ $ & � & � # � � � # � B h / & �

/ h B � $ / " $ h " � & / " # " / # / # B B

� B / # � � & $ & � & " $ B " " & / " # B # & � " � $ & � �

� & �

� � � / & / & � � � " $ & � & $ / �

� & � / $ $ / � # � " & / $ � & / / & " # / # & " /

� � # � � / h �

" $ / h " / #

� / � � � � h / / / & � � # � # $ / � � # � & # /

h / � & � & � � $ � / " # � " $ � $ � � � � & � & /

/ � � � & / / & � � � & � & / # � # � / " & $ & �

" # # / # � � # � & & � � h � � / & � � $ � � # � $ � B

" � � � & $ & / # # # & / #

Page 253: Goldrei.classic Set Theory for Guided Independent Study

( ) * + - / 0 ( * + 3 6 ( : 3 3 / 6 ) 6 : 3 ( : 6 3 : 0 0 6 * ( / (

0 ( * + (

* * : ) 6 3 * ( / ( / ( 0 /

( 3 ) ) ( * 3 3 * ( / ( ) - 3 0 - ) 3 0 6 : 6 - : 6 + ) * 0 + : 3

* 0 - 0 / 3 ( ( - 3 0 - ) 0 / 3 * ( / ( : ) + - : : 3 + 6 + 6 )

* + * ( 6

* * : ) : 6 3 * ( / ( ( / +

- ) ) ( 3 * + 3 * ( / ( ) - 3 * 0 - 0 + - - * ( / ( : + - : : 3 0 : 6 + -

+ ) 0 3 : 3 * - ) 0 / ( + / ( : 6 - + ) 3 * ( / ( ) 3 ( : 3 ( : )

+ ) 0 *

* * : ) 6 3 * ( / ( : ( 0 / + 6 0 / / ( ) - 3 0 - - ) / 6 ) 6 3 ( : 6 3 : 0 6

6 ( ( 6 0 / ) * 6 0 + - 3 ( / 6 + 3 : ( / 0 0 ) 0 3 * / ) - 3 : ) 3 + 3 ( ( / (

: 3 * 3 * : : 3 / : 6 + - ( + 0 ( ( : 6 + 3 + - - * + ( 3 0 + (

( 0 3 * ( 6 : ( 0 3 ( ) ) 6 + ( - 3 * + 0 / ( + 3 ( / ( ( / ( - : : 3

6 * * 3 ( + * 3 0 0 ) 0 ( : * 3 ( - - * + (

0 6 : 3 / 0 - ( 3 * 3 0 3 + 6 ( ( 6 0 6 ( 3 * ( / ( ) 6 0 + + 6 3 ( (

* 3 3 * ( ) 0 3 * / ( 6 : ( - / ( - 3 ( 3 0 ( + * 0 3 ( / 0 / + - 3 ( ) - ( ) ) + 6

* ( ( 3 * 0 ) 3 0 ) 0 ) ( * + 3 ( ( ) - ( ) 0 * - 0 / + - - 0 / - ) + 6

0 - ( 3 ( 3 * ( - ( + ) 0 / : * 3 + : * ( 6 3 / 0 ) * 0 3 + 3 3 : ) - (

0 + 0 6 + 3 : 0 6 ( ) * + - 6 0 0 3 * / 0 * 3 * ( ( 3 + : - ) 0 ) * + / 0 0 + )

3 * ( + / ( ( ) ) ( 6 3 : + - - 3 ( ) ( 3 * 0 ) ( 0 + / 0 0 3 / + 6 ) 6 : 3 ( : 6 : 0 6

6 ( - ( + : 3 / ( ( : ( / ( 6 3 + ) ( ) 0 / 3 ( - ( 3 0 ) ) : - ( + - ( 0

6 / + - - 0 / * ) 3 * ( 0 / ( * ( ) ( ) ) 0 / + ( ( 6 6

- 0 / : 6 + -

* : ( * 0 0 - ( ) ( / : 6 * 0 3 0 0 6 ) / * ( : / ( + 6 (

0 / ) ) 3 ( : 3 0 ( 6 ( + ) ( 0 0 / 6 + - )

: ) - : 3 0 / 6 + -

0 ( 3 * + ) + ) ( ( + ( : 3 ) ) ) ( 0 - ) 0 6 0 6 ( + )

) 0 3 * +

( - ( - : 3 + 3 : ) + - : : 3 / : - * ( / ( / ( 3 0 ) ( ) 3 0

6 ( / ( ( : 0 6 * ( 3 * ( / 6 0 3 3 * ( 6 ) + - : : 3

( 6 : 3 : 0 6 0 3 * ( 6 : ) + - : 3 * / ( - 3 0 ( / : ) (

* * ( ( / ) 3 * ( ( - : 3

Page 254: Goldrei.classic Set Theory for Guided Independent Study

� � � " � " � ) + � � + � 2 ) � � � ) 3 + �

2 + � � � ) 3 R 2 3 � � R 2 +

" � � � ) ) � 3 "

2 ) � " � � 3 ) " 3 + � 3 3 R " �

� ) ) � � � " � "

2 � � " � " � � �

R

) 2 ) � " � R

� ) R

� + � 3 2 " R � � 3 � 3 3 � R 2 " 2 � � R + 2

� � 3 � + � " R � 2 ) " � 3 2 3 � 3 � 2 R 2 3 "

+ ) ) " + � 3

� + � " R 3 � 3 + � ) " 3 + � �

) " + � 3 " � 3 2 � � ) R + ) ) � � R � �

) R � � + � � " � 3 + � 3 + " �

+ � 3 + " � � � � ) � R � 3 + � 2 ) � � � ) � � 2 � � � " � + � �

3 � � ) � 3 � 2 � 3 2 " � " � ) 3 ) � � 3 " " � R R

2 � 3 � + 3 " " � � � 3 + � " 2 � 3 3 2 3 � � ) 3

+ � " � ) ) " 3 ) " 2 � � R ) � � " � 2 ) " " 3 " 3 + � 2 " � � R

) ) 3 ) ) " � 2 � ) " � R � " + � � " +

R � � � 3 2 R � " � � " + � " " ) " 3 + � � 2 " ) + � " �

" ) 3 + ) � R 2 2 3 " R � � + 3 ) � � " 2 � R 2 "

� 3 " � R R " � ) � � � 3 + � 3

� " � � R ) 2 3 + 3 + � � � � ) R 2 3 " � � � 3 � 2 � � R + 2 +

) � 3 � ) + + R 2 2 3 � � 2 ) " � � � 3 � R 2 ) � � ) 3 ) � 3 2 2 �

" + R � � R � " � � 3 + � " R " 2 � 3 + � � R ) � " 3 " ) �

+ � 3 + � 3 + � � 2 � � � ) � � 2 � ) + � 3 + � 3 + � � ) 3 �

" 3 + + � � 3 ) " 3 � � � � " + " � R ) � 3 + � " � " �

" � 3 " � 3 � � � � � R ) 3 � � � )

" 2 3 2 " 2 � � R ) � ) " � 3 2 � ) 2 � " � � R ) " " 2 � ) � � � 3 "

� 3 3 " " 2 R ) � � � R " � � 2 3 3 " 3 � 3 � 2 � 2 R � ) 2 � �

� � " � R " � � 3 � � ) � � + � + � + � ) � � � 3 � ) �

Page 255: Goldrei.classic Set Theory for Guided Independent Study

� ! " # ' ( ' + ! � + , ( ' - . , " # ( " # , . ! ! + , . ! ! � - - , !

� � . ! " � , ! + ! ( " ! " # � . " � # � " # ' " ! + ! , ' ( � + , ( ' - . # # ' "

( ' " " # ! , . ( , ! ' . ! . ' - - - . � � " � . �

! ! ! . !

, # " # " , ' ( � + , ( ' -

, # � " "

, � " # , . ( � " , " � , ( ' -

. ( ! , " # ! + ( � + ' , , " � + , ( ' - , " . " ! ' . ! . � +

# � " # ' " , ( " . , . # ( ! ( � "

# !

� ( . , ! + " # ! � - ! ! ( " � , ( # , . , . ' ! - - � + ! + ! . ! " � , .

� + ! + , . � � + # , " � ' � + , ( ' - # � ! . , ( . ! " !

' + " ! " � � + ! +

# � " # ' " # ! � ! , ( ! , ! , " ! (

. " # ! + ! ' ( ! � + , ( ' - . # " # '

! " ! ( � + , ( ' - , #

# � " # ' " � . � ! " # ! ( " # ! + ! , . ' ( , !

# " # ' "

# � " # ' " # ! ! , . , ( ! ! . � ! . " ' "

! " ! ' ( � + ( ' - , " #

' # � # ' " " # ! + ! , . � ! � + , #

! " ! ' � ! # � " # " " # ! + ! , . . � ! . " # "

! " ' ( ! ' . ! � " ' " " # ! + ! , . . � ! . " # ' "

, " # . � ! . � " # " " ! + ! ' + ! � + # , #

! � . ! " # " ' + ! � + , ( - . - ! . . " ' ( # � " # ' � " # ' (

+ ! - ! . . " ( , ( " . ! + ! ' ! , " # ' (

! # ! + ! - " # ! � + ! ' ( ! + , . ! . ' ( " �

! . ' - ! ! + � (

! ! � + ( - . , # � " # " # ! + ! ' ! , ! � + , ' - . . #

" # ' " # ! + ! - - � .

' � " # "

! " ! " # ! ! # � # ' . ' . ! . . � � + ( ' -

. " . � + . " " # " , � + , ' ' ( " # ! ( . # � , . ( ! , " # ! +

( � + - ,

Page 256: Goldrei.classic Set Theory for Guided Independent Study

� � � ! � " # $ & $ " # $ � !

" " " " N & N

N W � # $

" & ! & " $ & N

! " "

" ! N # N N N # N N # N

� N N

" $ ! " W � " " N # � ! $ � " � N

N $

$ " N � $ � � # # " N ! !

W

$ $ N � $ N & & � N N N

N W $ " W # & N �

N W $ # # N � N ! & N ! � & &

$ � � # # " " N � " $ # $ ! N # ! � N ! � W � #

� ! W # � � # # N � " $ # # ! N N

W � # N N W # # # $ $ ! W �

# W N W #

# W � N N !

N W $ & N N W � ! # N

" & N " & W N # N # # N � ! $ # N $ ! � ! # � !

" � $ # # N & $ N # " $

$ ! � & � � ! " � !

" N W

� & " # " & N $ # " & " W & ! ! � N " N W � ! $ # " $ & "

& $ ! � $ ! & � & W " # & $ # N N # N

# N " � $ & & " W " # & # W # # " " � # � & N W � !

! � $ & � # � N # N " "

� N ! � ! N $ & $ # # # � N � ! #

N $ #

# $ � " W " # & � N # N W #

Page 257: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! & � 7

< � > A > & � ! & !

� � � � 7 A > 7 � A ! A

� & > � > � A A ! � < A ! 7 A � < 7 7 � � 7 & � ! 7

> < 7 7 ! e e < < 7 7 � A 7 � ! 7 � & � < � A < � ! � & �

> A ! 7 A 7

! � � 7 < A � ! �

A � > < � 7 ! ! & �

& > � > A > � � > < ! & > > � A � <

� � 7 � � & >

< 7

� & > < 7

< < � �

> ! 7 A < � A � ! 7 A A

� � 7 � � 7 ! �

& � & > & !

� 7 � � � A ! � �

� & !

& < � � & �

& > & ! ! & � > � 7 > �

Page 258: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � $ � � � � � ( � . $ � � � ( �

� � � � � � � � � � . � � ( � $ � � $ � � ( � � �

( � ( � � � $ � � ( $

� ( � � ( . � � � ( � $ � � �

( � � ( � � ( � ( � $ $ ( � � $ � ( $ � $ ( . $ � � $ �

( $ � � ( � $ � � � $ � ( ( ( � ( �

� � $

� �

� ( . ( � �

( . � � � � � $

( � � $ ( � .

� � (

( . � � .

( . �

$ ( . (

� ( ( ( � ( � $ �

(

( � $ � (

$ . � � � � �

( $

( � � $ � ( . � (

( � � $

� � . � � ( � � � .

$ ( $ � ( ( . � � � � � � � ( $ ( � � $ ( � � $

. � ( � � � � � � ( � ( ( ( ( �

. � � � � � � � � � ( � ( ( $ $ � ( ( �

Page 259: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � " � # � � � % # % " � � " � � � � � � " � # � � � % " � � � �

� � l # " " � # " # � � � " " " � � � % # � " � � " � " �

� � # % � � % � � " � � � " � � # % # � � � � � � � � � � � � # # #

� � � " # # # " � � � � � � � � � # # � � � � " � � #

� � � " � # " � � � � � � " % � � � � " � � � � � � � � # �

# % # % # # " � " � � % � " � � � � % � �

� � " � � � % " � # � � � � " # � # # � � # �

� � � � " � # � � # � � � " � � � # � � � � # % " � � # �

" # � � " " � � � � � � # % � �

� � � � � � � �

l � % � � l " # � � � � � # � � # � " � � � # # � � � �

# % " � � � " � � � � # � " � � # � � � " � " � � � �

# � # % # " � � � � � � � % # " � " � � � �

� � � # l # # � � # " � # � % # � � � � � � � � # %

% � � % � � � � � � l � " � " � � � � �

� � � # % " � � � " � # � � � �

� � � � � � � # � # # # # �

" � " l # % � # � # % # � � � # � %

" � � � � �

l # � � �

l # � l

� � � � � � � � � # % " �

� � % � " � � % # # # " # �

� � % � � "

# � � � % � � % # �

� � % � " � �

� # � � � � " # " � � #

� � � � % � � � # � � � � � " # # " # �

" # �

� � � � l " � � � # l � � # � % � # � # � � # % # " � � � �

l � # � � l " l # # % � � # " � � � � % � � � % # #

� � � # % " � �

� � � " �

# � � � � % � � # � # � � �

Page 260: Goldrei.classic Set Theory for Guided Independent Study

� � � � ! � " $ " � & " " $ � " & ! � ! � " $ " $ " � �

9 � � !

& ! � $ & " ! " $ " � $ " & � � N 9 � "

& � � N 9

& 9 " ! � ! " & " & $ � � N � !

� N � " $ " $ & $

� ! " ! " & $ N & $ & $ � N � " " &

" " � & $ � � N " 9 ! � � � & N N

" � � " � � N $ $ 9 � N

� " " ! � N & $ N N &

" & & � � " " � ! & � N � ! " � $ & " $ !

& " " $ � ! � &

� & " $ N � N �

� N " $ ! � � &

� $ � N & $ $ " 9 ! 9 & " " 9 N

! � " & � ! & � & & & $ � 9 $ � ! $ � � &

� � N & N & & $ � � N 9 & $ " & &

" " " " & & & & $ 9 & � $ " 9 N N

$ N " $ " & $ N 9 "

$ � ! " $ " ! " &

& $ " $ " " ! " "

" " N � � $ � " " & & $ � N 9 " �

" $ � N � � ! "

N �

� $ " " " � & ! & $ � N N �

Page 261: Goldrei.classic Set Theory for Guided Independent Study

� � � � � # � � ( � * # � � ( �

� * 8 9 9 � � ( ( � � # H �

� 9 � H � ( # 8 # 8 � 9 ( � #

� 8 9 9 � � # 9 � � ( # 9 # � 9 � � ( #

9 � 9

# �

8 ( � � � 9 � � � � �

� � ( ( � 9 8 ( H � ( � 8 � � # � � � �

� 8 � # H 8 � 8 ( 9 9 9 8 � ( � 9 � H # � #

� * � � ( 9 � # � � #

8 H 8 � ( � H � ( # # � H H � ( � (

� � * 8 � � ( � 9 8 � 9 # � �

9 � � #

8 * � # # � # # � ( 9 H 9 9 # � � ( # #

� # 8 H H 9 # # � (

� ( � H � #

� 8 9 � � �

� ( #

� � � � # �

� � � ( � H �

* � � � � �

9 8 # � � # # � �

9 � � � 9 � H �

� 8 H 8 � * � ( � ( # # (

( ( � (

� � ( � ( # �

� 9 ( � � H �

8 8 9 � # #

9 � 9 � H � # # � H � ( # �

8 # # � 8 H � ( � * 9 � � � � � � � �

Page 262: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � �

� , - 7 � � �

� � 7 � � T � 7 �

� - 7 � � � � � � 7 T � � �

� T T � � �

� - � � 7 � � 7 � � � � � � � � 7 � - T T � 7 � -

7 � 7 � 7 , � 7 � 7 � � � � � � � 7 � � - � � � � � �

7 � � 7 � � - � � 7 � - T � � � � � � �

� � 7 � 7 T - 7

, � � �

� - 7 � 7 T � - � 7 � � � � T �

- � 7 � � � , � � T T

� 7 - 7 � 7 T � � � �

� 7 � � �

� �

, � � - 7 7 - T T � � 7 , � � � - � , � �

, � � � � � - � � � � � � � � T - � � � � �

, - � - � 7 - T � � � 7 T � � T � - 7 � - T � � � 7

� � � T � � � � � T - � - � � � � � �

� � � � � � 7 T � � � T �

- 7 � T � � � � � - � � �

T - � 7 � - 7 T

� � T � � � � � T T 7 � � - T �

, - � � � � � � � T ,

7 � � � � � � � 7 � - T

- � � � � � 7 �

7 � � 7 � � � 7 � T

� � � � � � �

� �

T

� - � � � T T � � T 7

- � - 7 � � 7 - - 7 � � � 7

- � � � 7 T � � � � � 7 � -

7 � 7 � 7

- � �

� � T � 7 � � � � � T

Page 263: Goldrei.classic Set Theory for Guided Independent Study

� � � � � ! " " � � ! $ " $ � � ( ) $ � ! $ , � � " ! " � ! ,

$ ! " � � ! $ , �

� ( � , , � , " � ) ( � " � $ , " ! � ) � ! � " ) ! ( ! � � � ( � "

) " " ! ( ) ) ( � � $ , " � ! $ ( � � , , , � " , � " $ ,

" � � ! $ , , � $ � ) � � � � ! " � ( , � � ! � $ ,

� ! ( � � $ , ( " $ , , � � ! ( � " � $ , , ) � � � " , , "

! � ! ! � � " � ! , � $ � " � " , ,

, ( " � $ �

" � $ ) � � , � " ! " ) � � , " !

) � � � � $ � � , � ! ) " ) � ! � $ , " � � , ! ! � ) " � ,

� � " � � ! " � � ! $ , "

� ( $ ! " ! � ) " � � ! $ , $ ( , ) ) � ! � �

" � ! ( � � � $ ) � !

" ) $ � $ ! � � , " � , � " � �

" , , " � ( )

" ! � ! � ) � " � � ,

) � � � � ( $ ( ) � ! � ) � , " � � ! $ , ( $

� � � � � � ( $ ! � � � ! � � $ , " � , � ) $ )

" � ) � " � � " � � � ! ) , � !

$ ( � ) � ) � , $ ( ) � , " " ) $ ( � �

� " � ) � � (

� $ ! � � ! � ) � , " ! ( " ) ) � $ " ! , �

� � � ! � ) $ , " � ! $ , ( , � , $ � ( � � $ , ( � ! � $ ! � " � � ( � � � ! � � � "

� ! ! � � " � � ! $ , ( � ( � $ , , ) ) � ) ) " ) � � " ! ) � � ( � ) � � $ ) " �

Page 264: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � $ � � % � � % $ � � � � - � �

i � � � - � - � � % � � i � i $ �

� % $ � �

� i � � � - $ � - � � - $ � � $ � � % � $ � � � % $ � � � �

- � % � � � i � � % � $ � i � � � % - � � � �

- - � � % � � i - � � % � % � � � - � i � � � � � � �

� � � � - � i � � i $ - � $ � � - $ � % � $ � - � � � � % �

� � � � � � � � � - % � $ - � � � � � � $ �

i �

� � � i � � � i � � �

� � � � � i i � i � � i $ �

� � � - � � �

$ � $ � - $ � � � $

� � � i � % � % �

i � � � $ � � - � $ � - � i � � i $ � � � � � � � � �

i $ � � � � $ � � � $ � � % � � � � % � � � � �

� $

� $ i � - � $ � i � % - � � $ �

� % � i � � � - � � � � i � - � � � % � i � � � � � $ � � � � �

i � � - � - � i � i $

� $ - - � % - �

� � i � i � � � % � $ % � $ $ � � �

$ � � $ % � $ i

� � � - � � � % $ � � � � $ � � � % - � $ �

i � i � � � % % - � � � � - � i � �

Page 265: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � " $ " $ � � $ ( , � � " , ( � � � , , " � 5 � � (

; ( � 5 � " ( ; � ( � , � ( ; ( � " � � ( � " , ( � (

; � � ( � � � $ " � $ � � , " ; � , � 5 � � � ( � " " � , � 5 � � " � 5 5 (

, ( ( " � $ " � � � $ � � � ( , " � " � ( � $ � � ,

� ( � , $ ( ,

� �

� " � $ � $ � � � ( � ; � " � $ � � � , " ; � , � 5

( � ( " � � � � � � � ( � " ; " � , � 5 � " , � � � � ( � $ (

, " ; 5 � � � , � , � � � 5 " ; � , � 5 $ � � " ( � , , � � ( � " � $ � �

" ; ( ( , � 5 $ � � ( � , � $ � � " � $ � � � , " , ( �

( " 5 ( � 5 � " � , � 5 � " � $ ( , � � " ; 5 5

( $ � ( � � , ( ; " , " ; $ ( ( � � "

� $ �

5

� � � � 5 � �

� � $ � � " 5 ( , � $ � � � $ � � � " � � � 5 ( $ � � , , " �

( � 5 � � � $ " , 5 5 ( � ( � � $ ( " � � � � 5 � � � � � � � � ( � � " " ; � , � 5

� �

( � $ ( � ( � (

� " $ � $ �

5 5 $ ; ( ( " � , � ; ( � , � � � 5 " ; � , � 5 � � , ( , � " , � �

� $ ( 5 ( � � , � � " ; � , 5 ; ( � � ( ; � $ � , ( � � $ $ � (

� ( � ; (

$ $ ( � " 5 ( ( 5 ( � ; � ( $ ( � ( ; ( , " � � � 5 ( � � ( " ; � ; , � 5 �

� $ ( , � ( " ; � , � 5 � $ ( " ; ( ( ( � , , � ( " � � 5 � ( � , ( " �

$ " ( , � 5 " ; , � 5 $ ( , ( " � " ( , $ � ( ; ( � " � ( � � � ,

$ � � ( ( , � � $ ( � ( " ; � ; , � 5 � , � , , � ( � ( " ; " ; � , 5 � ( � (

, " , � " � � " , � " � � $ � ( ( ,

( � " ; � , � 5 $ ( " ; � � � , � ( (

" 5 "

� ( � �

" , , � ( � � ( 5 ( � ( � , ( ; " � � �

Page 266: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � �

� � � � � ' � � � ' � � � � � � ' � ' � � � � ' �

� ' � � � � � � � � � � � � �

� � � � � � ' � � � � � � � � � � � � � � ' �

' � � � �

' � � � � � � � � ' ' � � � � � � � � � �

� � � � � � � � ' � � � ' � � � � � � � � '

' ' � � ' � � � � ' � � � � � ' ' � � �

� � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � ' '

� � � � � � � � � � � � � � � � � � � ' � � � � � � � � � �

� � � � ' � � ' � � � � � � � �

� � � � � � � � � � � � � � � � � ' � � � � � � ' � �

� � � � � ' � � � � '

� � � � � � ' � � � � � ' � � � � � � � � � � '

� � � � � � � � � � � � � � � � � � �

� � ' � � � ' � � � � � � � � ' � � � � � � � � '

� � � � � � � � � � ' � � � � � � � � � � �

� � �

� � � � � � � ' � � � � � � �

� � � � � � � � � � � ' � � � � � �

� � �

� � � ' � � ' � � � � � � � � � � � � �

' ' ' � � � � ' � � � � � � � �

� � � � � � � � � � � � ' � ' � � � � ' � � � � �

� � � � � � � � � � � � � � � � ' ' ' � � � � � � ' �

� � � � � ' � � � � ' � � ' � � � �

� � � � ' � � � � � � � � � ' � � � � � � �

' � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � ' � � � � � � �

� � � � � ' � ' � � � � � � � �

Page 267: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � ! � � � � � � � � � � ! � �

� � � � � � � � � K � � � K � � � K

� � �

� �

� � � K � � � �

K ! � � K � � � �

� �

� � � K � � K �

� K � � � � � ! K � � K � K � K � � � � � � K � � �

� � � � � K � � � K � � � � � � � � � � � K � K � � � � � �

� � �

� � � ! � � � � � �

� � � � � � � ! � � � � K � � � � K K � � � � � ! � � � � �

� � � � �

� ! � � � � � K � � � � � � � K � ! � K � � � K � � � � �

� � � � � ! � � � ! � � � � ! � K �

� � � ! K � � � � � � � � � � � � � � � K � � K �

� � � � � � � � � � � � � K ! � � � � � � � � � K � � � ! K �

� � � � � � � � � K � � � � � � ! � � ! � � K � � � ! �

K � � � � � ! � � � � � � � ! � ! � � � ! �

� � � � � � K ! K � � � � � � � � � ! � � � � � K � � � K � �

� � � � � � �

� � � � K � � � � � �

� � � � � � K � � � � ! � � � ! � � � � �

� K ! � � � � K � � � � � � � � � � � � � � � � K

� � � � � � ! � � K

� � � ! � � ! K � � ! � � � � K � � � � � �

Page 268: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � � � � � � � ! � � � � � � ! � �

� � � �

� � � � � � � � � � �

� � � � � � � � ! � � � � � � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � ! � ! � � � � � ! �

� � � � � ! � � � � � ! � � � � � � � � � � � ! � � � ! � � � � �

� � � � � � ! � � � � ! � � � � ! � � � � � � � � � �

� !

� � � � � � � � � � � �

� � � � � � � ! � � ! � � � � � ! � � �

� � � � � � � ! � � � � � � � ! � � ! � � � �

� � � � � � � � � � � � � � � � � � � � � � � � � � � !

� � � � � � � � � � � � � � ! � � � � � � � � � � � � � � � ! � � � � � � � �

� � �

� �

� �

� � ! �

� ! � � � � � � � � � � � � � � � � ! � � � � � � �

� � � �

� � � � � � � � ! � � � � � � ! �

� � � � � � � � � � � � � � � � � � �

� � ! � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � � � � �

� � � � � � � � � ! � � � � � � � � � � � �

� � � � � � � ! � � � � � � � � � � � �

� ! � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ! �

� � � � � �

� �

� � � � ! � � � ! � � � �

� � � � � � � � ! � � � � � � ! � � � �

� � � � � ! � � � � � � � ! � � � ! �

� � � � � � � � � � � � ! � � � � � �

� � � �

� � � � � � � � � �

Page 269: Goldrei.classic Set Theory for Guided Independent Study

� � � ! " # $ & & � ! $ " , � � " # $ 0 & & , & $ $ " # $ , " # , " � 9 � � � "

& 0 � ! , � " & 9 # & " # "

# $ & $ � & ! � " # " # $ 0 $ 9 � " & $ � 9 $ & $ " & ! 0 & � $ 9 ,

9 � � $ , � " $ 0 ! , � $ 0 & & " , " $ � $ , $ " # $ $ , � � 9 & 0 & , 9 , !

$ $ 0 9 $ $ $ $ 9 # , � � � $ , $ $ 0 & & & " # $ , ! , � & & 9 0 $ 9 � " " & $ 0 9 $

, & $ & 0 " # $ � " � � � , " & ! & � $ 0 $ ! " 9 9 , 9 " 0 , � " & 0 $ $ 0 � 9 $

& 0 &

9 $ # $ & $ ! " # $ 0 $ � " & $ 0 9 $ " & 9 # & " , "

# & " , "

& � " # $ 0 $ & � ! " & $ & 0 � ! ! " $ , � ! , � ! $ 9 # $ , � " � &

, ! � � , " & ! 9 � ! , 9 � 9 � " � � ! " # $ ! $ " # , " $ 0 $ 9 # , � � 9 $ $

# & " # $ � & & # & $ 0 $ 9 & � $ , ! & " # $ 9 $ 9 9 0 & ! � ! 0 � ! , � 9

, ! & ! ! $ # $ 0 � " # & 0 � ! , � � ! , " � � 0 " # $ 9 , 9 � ! , ! & 0 9

& � ! # $ & 0

$ " $ , ! ! � � � & 0 ! � , ! # & " # , "

# & " # � 9 , ! ! " � , � & � ! , � , ! , 0 $ 9 $ " 9 # # ,

" # $ ! & 0 � ! 9 9 & $ " # , 9 & ! $ & ! $

, ! $ & 9 & $

, !

, ! $ & 0 9 & $

9 & " # " , ! $ # & " # , " & " # , , ! $ $ � � & 0 $ 0 $

& # , " # $ $ $ � & 0 , � 9 , ! # " # " , !

& $ & # $ " # " � 9 ! � " � , ! # $ 0 $ � " & $ 0 $

Page 270: Goldrei.classic Set Theory for Guided Independent Study

" $ " & ) & - & - & & ) 1 - 6 " " 9 : " $ 9 ) - & $ " C "

6 " & : ) : " ) " 6 9 " 6 9 9 " ) 1 & 6 1 9 - & " - 6 " $ "

& 9 $ " & 9 $ 1 & : & 9 C " " & ) & 6 " " $ & C "

1 " - 1 $ " ) - 6 & 1 " " ) 9 " 1 & C 9 " 6 : & $ & 9

& $ " $ $ & 1 & - : ) ) " & 6 " 1 " : 1 " 1 6 $ & C " - 6 " & & 6 & "

9 1 & C C " " ) 6 : 6 9 " : " - $ : C 9 6 - 6 9 & $

& $ C : : ) 9 $ 9 C 1 9 ) & 9 C 1

$ 1 & " - $ " & $ " C " 1 & 1 " & 9 ) - & $ 1 " 1 6 6 9 "

& " 6

6 " 9 ) ) 9 $ 9 & ) 9 C 1

" 9 " $ " $ " 9 ) " 1 6 9 - ) 1

" & - 9 ) - " ) " ) $ " 9 C ) " $ 1

" " 9 9 1 9 ) & : ) - 1 & ) - 6 " $ 9 C ) " : ) - 6 " C "

C & 6 - " 1 " " $ : $ " 1 1 " " 1 C 1 " " $ & " & $ 9 ) " 6

$ " 1 " - 6 " 6 $ " " $ ) 1 - " : - 6 " & " - " C ) 6 9 & - 6 & &

& 9 $ " 6 " : & 1 " 1 : 6 & 6 : ) " 6 9 " "

6 " & 9 6 9 " 1 " 9 & 9 ) ) ) " " - 9 C

9 $ " & & 1 " " - " C " : " - - & 1 9 : - 6 9 - - : 1

6 " 1 1 6 6 9 $ & " $ : 1 " " & & ) 9 :

" " & " " 6 6 " 6 $ - 6 9 " 6 " " C & " 1

- 6 " 1 & C - ) ) " $ ) 9 " & " ) "

& " 6 " & " & ) - 1 " ) " & 6 1 & "

) - 6 " - 6 " " - & ) " " - 6 " 1 " & " ) 9 " & "

& ) 9 " $ " & " - 6 6 & & 9 9 6 " "

& 6 6 & & " " " " " C & $ " "

) 6 " " - " - & ) & - 6 1 6 9 " " 1 6 & & 9 6 " " 9 & ) 1 : " " "

6 " 6 6 " $ " 9 - " 9 1 "

6 9 " " " " C C $ " $ " ) - 6 " 6 : " - ) " 1 6 9 C & & 9

6 : 9 & ) : - 6 "

" - " 6 & & & & : - ) $ 1 ) " " ) 9 ) " 1 - - 6 "

9 " 6 " "

6 6 - 6 6 1 & ) - : & 6 6 & $ $ "

) " " 6 1 & ) ) : " " 1

1 - 6 " 9 " C & "

6 " 9 ) ) : " 1 " - & " ) " & 1 6 6 " $ ) & $

) 6 " ) 1 - & - 6 6 9 - $ & " 6 9 C C 1 - 6 " " : 1

" 6 " 1 ) 6 6 - 6 & " - 6 " " "

Page 271: Goldrei.classic Set Theory for Guided Independent Study

0 1 0 3 4 7 9 0 4 7 4 0 B C E G 0 B G 0 0 4 0 1 0

0 B C 4 1 0 9 0 B 9 C B C 3 7 4 0 0 3 4 1 0 1 4 E

G 0 7 9 4 G C 4 G 4 0 E G 9 0 B C E 0 9 G 7 3 4 4 7 1 4 4 4 B

E 3 G 0 B B 7 3 9 4 4 B G 0 0 7 3 0 E 0 G 0 4 4 G 1 0 0 0 E B 3

1 0 7 0 0 7 7 3 1 C 0

4 0 B G 7 9 E 4 E 0 9 0 B G 0 E G 0 0 C 9 7 E 4 E

G B G 7 B 4 3 E E 0 B 9 9 0 4

0 1 0 B 0 0 4 0

0 B 0 7 E 4 E 0 B 9 0 B B 3 4 1 0 0 B G G 0 1 4 0 1 B

0 0 G E E 0 4 E E G E 4 E E 7 B 3 0

4 7 3 0 B

9 G 3 4 B G 0 1 4 G B B G 0 9 0 C 0 0 0 3 0 B C E 4 9 0

7 1 0 9 7 0 0 9 G 3 C 0 0 7 7 0 4 4 G B

B 0 B 4 4 B G 0 4 4 G 7 0 9 9 4 9 1 G 4 7 1

G 7 G B 3 4 B 1 B B G 0 0 B 0 0 0

0 4 0 0 9 0 B B G 0 0 C 7 0 4 4 E E 9 4 9 0 B G B

C 0 9 0 B G 4 G B G 9 7 E 4 E

9 G B G 7 4 1 3 3 4 9 9 0 B 9 4 E B G 0 0

7 0 3 1 4 E 4 B 4 0 E 0 0

0 0 7 1 0 4 E G 0 E 0 7 G E B 0 4 4 E B 7 E 9 7

0 0 0 0 E B E 0 G 0 G 0 E 0 7 G E B 4 E

E

0 G 7 G 0 B G 0 4 E G 0 1 0

G 0 0 0 C E 0 9 B G G 4 0

G 0 4 0 0 0 7 0 9 4 G G

Page 272: Goldrei.classic Set Theory for Guided Independent Study

! " % % % " * " - . ! " " 0 % 3 " 3 * 5 3 * 5 * 0 "

" " 5 5 % 5 " . " % " 5 " D " 0 0 % 3 " % 5 " - ! D 0 - .

% 5 - . 3 D 5 . 5 % D 5 D 3 % 3 D . " 0 ! % - " 5 % 3 * 5 D 0 . * 0

. ! " 5 " " . 3 ! . % D . % " 5 " " 0 0 % " 3 % 5 " 0 % * 5 . ! "

" 0 " % 5 . ! " % 3 * 5 0

. ! % % " ! % - " . ! " % 5 D 0 ! " D 5 - " 3 % " - % D 3 . % - . ! " % "

! " % 3 " * 5 " . % 5 ! * - * " - - D 5 % * 5 0 - ! . D .

! % " D 5 5 % . - " ! D . . ! - " D 5 - . D " D " . ! " " " 0 * 5

. D . - % " " - " * - * " 3 . D ! - 0 - ! % * 5 % 3 % 5 - . 3 . * %

" % * 5 ! " % 0 0 % * 5 D " - ! 0 3 . % " " % 5 " 5 " 5 . % 5 3 %

. % . 5 - 5 . " 3 " 3 * % 5 D - " - . 5 % . % " 0 " " 5 . - %

- - " * " - " 3 " ! " " 0 0 % ! " % D * 5 ! * -

" D - . ! . " - ! 0 0 5 " " - % " - " . 5 % . * 5 % 3 . ! " 3 " * 5 " 0 " " 5 . - %

% D . % D 5 % D * D *

! " " 0 0 . 5 3 " . % 0 0 " . % " 5 " % 3 " !

. % " % " % . " " " " 5 . - % % . D 0 3 " D - " D - - % "

" D " 0 " . % 5 % . * 5 D 5 " % % % - " % 5 " % . ! " - " " " " 5 . - "

- ! 0 " " D ! % * " 5 . * % 5 % 5 5 % 5 " . - - " . - % % D - - * 5 D -

" . . " % . ! * - . ! " % 3 " . D " D % * " 5 . * % 5

. . % D 0 0 5 " - - " . -

" . 0 . ! " * 5 3 " * " 5 . % % 3 % 5 - . 3 . * % 5 . % " . ! " 3 " " D

5 % % 5 . 3 D - 5 * " 3 " - * % 5 % 0 0 % -

D 5 " D 5 "

% . ! " * - "

" - ! 0 0 " D % . % " 0 % * . . % % 0 " . " . ! " 3 % % D - " * - " " 0 %

% . ! % ! . ! " " . 0 - % " " % . " % % % " " % 0 0 % -

D ! % . ! . * " . ! " 5 % " D 0 - - % . ! . % . 3 " 5 * . !

. " 5

" . ! " 5 " D 5 5 " % D 3 D . ! D . - % 5 " % 5

D * 5 . ! . - " - . ! D " D 5 " " . " . ! "

" 5 . % ! . .

! % . ! . - D " . * % " . " " 5

% * . % " 5 " D " 0 % " % 5

" . - 5 % 5 . % . ! " * " 5 " % " " % . % - % . D

0 " - " 3 - " * 5 " . % 5 % 0 0 " 5 " " . % - ! % ! " " * -

Page 273: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � �

c � � e e c e � � k � � � e l m � � �

� � � � m m � � � � � k k e l � l � � � � k k � � � � e � �

� � l � e l � l � � c � c � l e � l � � c � � � � c � � m � m c � � m � l e m

m � c � � e � e � � e m � � � m � � � k � � � e m � k � k � � m �

l � � � k k � � e � � � � � � � e � � k m � m � e l

� � � m � l � m m � � � l � l � � m l e � m � e � � � k � � � � e

� m � e l � e � e l k k m e l � � k �

k � m � � l e l � � � � e � k k e l m � m � �

� � m � � � � k � e l � � k � e m � � � l � e � � e � � e � �

� � k k � m l � � � � e � � m � � e e e m � � e l m � � � e

� � � � � m m e � � m � l e � � m � m e l � l � � � m � m � � e m

e m � m � e l e � � k k � m e � e � � e m � l e m � � �

e � e � c � � k � � m � e � � � � k � k � � m � e

� � � � k � c � � m l � � � m � � l � � l � e l e l � � � � � � � m e l � l � � m

l � � � � l � e � � e � � e � � � � m e m k � � k

� � � � e � m e � � � m e l � � � e e m � � � � � m e k k � k � m � �

� m l � � l m � l � � e l � l � m � � m � � k k k k � � � � � �

e � c � �

e � � � � � � � k m e � m k � �

� l � � � � � � � e � � m � � e m m m e m � � e �

� l k k m e m � e

� � k � e � m � m � � � � l m � � c � k k m � m e � � m

e � � � k � � � � e l � m e � e � � k � � � � m m e

c � � � � e e m � � � m � � m � � � m � e m � k e � � e � m � �

m � � e m e k k e �

e c �

e l � k � � ��

e � l �

� � e � � � l � � c � e � � � � m � e � � e m � � e m � � � l � m � m

� � � m � � � m � e � m e � � � �

� m e m k � c � l � � � e � �

� m

e m � l � � e �

e l � k k

Page 274: Goldrei.classic Set Theory for Guided Independent Study

& ' ) * ' ' 4 5 6 ) * ' > & ' ? ) 5 5 5 )

' > ? > Q ) 4 Q ? ' 4 Q 4 ' Q 5 ' 6 > 5 ' ' 6 ' Q

> & ) ? ' & Q & ' & 4 5 4 ) ) ) ? * Q ' 6 ' 6

& ' * ' 4 ? ' ) ) ? ' ) ' > Q ) ? * ) ) 6 ) > & 4 > ' > & 4 '

) * ) '

> ? ? ) ? ' ? ? ) > ' 6 ) > & >

Q 6 ' > & ' * 4 6 ' 6 ) * 6 ) ' ) 4 Q Q 6 ) > ' ) * & 4 >

? ' 6 ) > 6 ) ' 6 ? ' 4 5 Q > > ' ) ' ' 5 ' ' ? > )

? > 6 > ' > & 4 > > ' ' > Q ' & ' 6 ' > ) 4 5 5 ' 5 ' ' ? 6

) * ' ' > & Q ? > & ) ' * Q 6 ' > Q 6 ) 5 ' 4 ? 4 >

6 Q 4 ' ' ? > ) & Q & 6 4 > ' 4 ' 5 ) ) Q ? *

& > ) 5 & 4 ' ? Q 4 6 4 5 Q Q ) * Q ? 4 5 & ) ' ? > & 4

' ' ) * 4 4 ? > & 4 ) > > ' * ' 6 ? )

Q & ) 4 5 > 6 ) ' * ' 6 ' * Q ? 4 ? ) ) * 6 ' > '

) Q ? Q 6 5 ' 4 * 5 ) * ' * ' 6 ) > & 4 & ' ' ' ? '

) Q & 4 Q ? ' Q ) ? Q Q ) ? ) ' ) ? 5 5 ' Q Q * ) )

& ' * ' ' 6 ' Q > & Q ? ? ' * ) ? 6 Q ? 4 > Q

) ? ' ) > ' ' 5 ' ' ? > ) > & ' 4 Q ? Q ' 5 ) 6 ) ' & ' ?

4 6 4 Q Q > ) Q 4 5 5 ) 4 ' 6 > 4 > Q 6 4 5 ) ? 5 ' ' ? )

> ' ) ? * Q ) ? ) 4 ? > ' ' & 4 )

? ) ? ' ) ? > ) 5 5 ) * ) & Q 6 ) Q > Q ) ? & 4 > ) ' 6

? ) ' 4 ) * 4 ? ) 4 > Q ? * > & ' *

) * > ) ? & 6 4 ? ) ' 4 5 Q ) Q ? 5

' 4 ' 5 ' > & > ' ) ? Q ) ? Q 6 Q ? ' ' & ' ) * ) ) '

4 ? & 4 Q 6 Q 4 ' 5 ' ' ? ) 4 Q '

) * & ' ) 5 5 ) Q ? ' 4 Q 5 ) * ) ) ) ) ' ) ? Q > ' *

? ) * ' 5 6 6 ) & ) > & Q ? Q > ?

Q ? ' * 4 ? ? > ' ? > Q ) ? ) ) * '

' ? & ' ' 5 6 ) > ' * ' ) Q ) ? ) > & ' ' Q 4 5 ' ? ) > )

? ' ' ? ) 5 ' ) ' ' ' Q ) ) ? > ' & ' ) ) * Q ? 5

> ' 6 & 5 ' 4 5 ' ) ? Q * ' 6 * ' > Q ? ) 4 ? > ) * ) Q

Q > ? > ' ) ' * Q ) ) 6 ' > Q ? ) Q ? & ' Q ) ) ) Q

' 4 ' ? ' > & ' * Q ? 4 5 ? 4 ) 4 ? Q ? Q > ' ? ) > & >

& ? ' 4 * ' Q ' * 5 ' ' 4 ? 4 ' ) * 4 Q > > Q Q >

> & ' ' & 4 5 5 4 ) 6 ' ' > & ' ' ' ? > ) & Q & > ) 6 ' ' ) 4 * Q ? 5 6

> 4 5 5 * ' ' & ' 4 Q ) ) ) Q ' & ) ? 4 ? ) & Q 6 5

4 Q ? & Q & ) 5 5 ) * ) ' > 4 5 5 ' Q 4 ' ? > > )

Page 275: Goldrei.classic Set Theory for Guided Independent Study

� � � � � ! " ! # � ! � � & � # " � " & � � � " & 0 & & � � � " &

! � � � � 0 � � � � # � & " ? � 0 # ! # � � � � ! 0 � � & � � �

! # � � � � 0 � � � � ! � �

! # � " � � " � & � ! � � ! " & � " & ! # � " 0 & " ? � 0 ! &

� � " � ! # " ? ? � � � � 0 � ! � � � ? � " � � � � ! � ! # ? � �

! � � # � " � 0 & " � & " � ? " & � � & " ? " � � ? ? � � 0 & # 0 #

� 0 " ? � & ! � & � � ? ? � � � � � # � � " � � ! # � & � � � ? � 0 �

! � # # � � � � � ? ? � � � � � � & " � � � � � 0 � � � 0 " & � 0 ? ? � � 0

� & � � # � ! # � " � 0 & " ? � # � � 0 � & " ! 0 � ! # � ? � " ! � #

� � 0 & " ? # 0 � 0 � � � � � & ! " ? ? ! # � � & � & ! � � � 0 � � � � ! � & # 0 #

" � # " ! ! # � � 0 � " & & 0 ! 0 " ? � � 0 & " ? 0 ! # � " & � " � ! # 0 � " � "

� � �

0 ? � � ! # " ! � � � � & & 0 ! � � � ! # � � � 0 � " & � ! " !

# 0 " ? � � ! � � & � � " � 0 & " ? � " � � ? ? � �

� ! � " & & � � � ! & # � � � �

0 0 ! � & " " � 0 � ! # � 0 & 0 0 " ? � � & " ? � �

# #

# � � � � � & � � ! � � ! # � 0 � " 0 " ? � � 0 & �

0 � " ! � � 0 & 0 0 ? � & "

� � � " ? � & ! ?

0 � & � � 0 & "

� � ! � � � � � � " ! # ! # � � ! 0

" �

& � " ! � � & � � � & � � ! # � � � � & 0 � & � � ! # " !

� � � �

& # � & � ! � # � � � � � ! � " 0 # ! � � " � � � 0 � � � " � � ! � � # � ! # "

# � � � � & ? � & � " � & " ? & � � � � " # � 0 �

# � ! # " ! � � " & � � " � " & � ? 0

� ! # " � & ?

Page 276: Goldrei.classic Set Theory for Guided Independent Study

+ . + 5 6 7

6 . 5 7 5 7 j j . 6 m j j 7 . . s t 5 6 6 s t m . 7 s t

. 7 j m 7 s 6 7 . 7 j 7 � + t 7 + t s t � 6 . 7 6

7 j + 7 m . 5 t m . 5 m + 7 7 j + + t 6 7 . . 7 7 5

7 7 t j 7 s + t 7 m 6 + + t m . 7 + j 7 6 m j 7 + 5 j t � . 7

� 6 + � � + t 5 6 7 � + 7 j 7 m . 7 6 5 � 7 m 7 j 7 5 s + t

m . 7 � 5 6 m . 7 t � 6 + 7 7 � + j 7 j 7 m + j 5 j t 5 � 6 t

7 5 t s t s � + j t � 6 . 7 7 � 5 m s + t s . 6 m 7 7 6 j 7 7 5 �

6 t t . m s 5 t + t s + 7 6 . 5 m 5 7 m . 7 6 5 5 6

j j 7 j t m . 5 j s 5 � 6

7 5 j 5 � 7 t 7 6 . � 5 . 5 7 m � . 7

s . 7 s . 7 + 7 j + 7 +

+ j 7 + t 7 + m . 7 j + 7 j m s 7 6 + + + j s t s 6 m . 7 � s t 7 5 m j + 7 j m . .

s t m 7 j 6 + . 7 5 6 6 + s 5 7 7 5 + j 7 j 5 t 7 j s m 7 t 6

+ � + j

. s 6 m . 5 m j 7 6 + � 7 5 7 s 7 t + � 7 + 7 j j + . 5 m 7 j 5 6

� +

s 7 6 . 7 . + m + j s 7 7 . 5 m + j 5 t 6 7

+

t 7 j + 6 7 6 m . 7 5 m . 5 m 5 6 7 7 7 m s 7 7 j + 6 +

m 6 5 j t t 7 � m . 7 + j s t � 5 m . 7 � t 7 5 j s m + m . 7 6 � + j 7

+ t m 7 + j t 7 s m . 7 j 5 j + j 5 j 5 m . + m

+ 6 6 7 5 s � 7 m 6 6 + 7 . 5 m 5 j 5 . . s � 7 6 . 5 m

5 j 5 j . 7 6

5 j

+ m .

7 m 7 t + t 7 m 6 7 5 t 5 6 6 7 + m . 5 m m . 7 j 7 s 6 t 7 � 7 7 t m

6 . m . 5 + j � � + 7 6 m . 7 j 7 6 � m 6 m s � . + � s s 6 5

j + 7 j

Page 277: Goldrei.classic Set Theory for Guided Independent Study

� � � ! & � � � ! � . / & � 1 ! . / & � � & � ! / � ! 4 � / . . � � �

& � ! . / & � � ! � . < � 1 � � 4 . � & ! � 1 . 1 � 4 ! . � / � & � 1 ! / 4 / ! 1 ! �

! . ! . � 1 & � 1 . � � & . < / ! 1 / . < � . /

� � � ! � � . � ! � � � � � . � & � 1 . / & � &

! � � � � & � . 1 / � 4

� � 1 � � & 4 4 & �

! ! ! 1

! ! ! 1

1 !

& & � � . � �

1 1

� �

! ! 1 !

� ! � � 4 � &

! 1 !

� � & . . / & � & 1 1 / � 4 & � � / ! / & � / � & � � / ! 4 & � � / � �

1 � ! 1 / � � � . < � � � . � � � 4 � ! 1 ! 4 � 1 � & 1 & 1 / � ! 4 � & � �

� ! � . < � � . & � . & � � 1 & . & & 1 / ! 4 & . / � . / / &

. < � ! 1 / � & � � / 4 / � � & � ! . / . / 4 4 � 4 � 1 .

& � . � < / & . � / � . ! 1 � / ! � � ! . / & � & 1 / � � ! 4

/ � & 4 / � & . < � 1 / � 4 � . � & 4 & 4 4 �

. < � ! / � � & � � . / 4

< � 1 � � 4 & � . / & � . ! / � ! � 4 / 4 / � . < � & 1 / � ! 4 � !

. < ! . . < � ! . / & � ! � 4 . / 4 / ! / & � & ! 1 / � ! 4 / 4 4 � 1 < ! 1 1 / � / 4

� &

� � � ! 4 � . � � . . 4 ! � & � / / / � / .

! 1 1

! ! 1

& 1 ! � & & 1 � ! 4 � . � / 1 & 1 / � ! 4 � / � � / � � 1 & � /

� ! 1 / 4 ! 1 . < / 1 � � 4 < & 4 < � � ! � 1 �

/ ! 4 � ! . 4 � ! & � � & � / / � . � ! 4 � � . & � � & ! 1 ! �

1 / � ! � & 1 & � ! � < � & � 1 / � � . � 1 ! � & 1 ! � �

Page 278: Goldrei.classic Set Theory for Guided Independent Study

� ! # ! % � * ! 1 % 6 * % 1 6 1 6 � 8 1 ! # 8 6 * 1 % 9 � ! # 6

% � 8 � * 1 8 * 6 * % 1

8 8 � 8 �

1 6 9 � ! # ! % * 8 1 % 6 ! 8 6 * % 1 � 8 1 # 8 * %

6 9 * 1 6 � ! # 6 % � 8 � 1 8 # * # * 6 * %

9 ! # % 6 9 * ! 6 9 % � 1 � * 6 1 * # 1 6 � % � * 1 #

! * % � 8 1 8 * 1 ! 6 # ! % * * ! * 1 1 * 6

9 % 9 8 % !

! * 1 8

8 � �

� % �

9 * ! � ! # 6 8 1 * 1 6 * * * # *

% ! % 6 * % 1 * % * 9 % 1 * * 6 * 6

! % % ! % � 6 � * 8 % �

% ! 6 � # 6 % % �

* 1

� ! # 6 % � * 1 � * 6 1 1 % 8 � * # � * 6 6 *

8 � 8 � 8 % 6 � # 6 % 8 � * 1 8 #

8 � * 6 9 * # * 6 * % 1 % # # � � # * � � ! # ! % 6 * 1 * 6 %

* 1 ! * # # � # % 6 8 % * ! 1 6 #

6 9 8 % � 8 # # ! 6 � % * 8 # * %

% # # 8 � * 1 #

8 % # #

% � # #

8 ! 6 % 8 * 1 8 # ! 8 8 * % 1 # * 6 8 � 1 % ! 6 # 6

% # # % ! 8 1 * * # # 6 8 6 8 1 % % 6 6 6 * # * * ! 1 % # * 1

1 ! # # ! 6 * * 1 1 6 % � * 6 * 1 % � * 8 # % ! * 1 8

* 6 6 *

Page 279: Goldrei.classic Set Theory for Guided Independent Study

� � ! % ! & ! . 1 3 . 1 � 7 ! . ! 1 7 8 & 7 ! 8 % % 8 3 .

7 8 �

3

3 3

3

3 3

8 ! 7 ! 8 ! . & ! ! ! 8 ! 3 8 .

3 ! � ! & 7 ! ° ! ° ! ! 1 & ° 3 ! ° ! % & � 1 8 & 3 . 1 % . ! 1 % ! � 8 .

� ! & 7 8 � ! 1 ° 3 � . ° 8 ! ° 3 ! 8 ! & 8 . 1 � 8 % % 8 �

� � ! 1 . ! & ! 1 . 1 ° 3 � 3 & 7 7 ! .

3

3 3 3

3

8 ! 7 ! ° !

° 8 ! & 7 ! 8 8 3 . ! � % & � 1 . 1 % 1 ° 7 ! & 3 1 � . 7 ! ! 1 .

! & ! 1 . �

7 ! 3 . � 7 8 & 7 1 &

! 7 ! % ° ! 1 ° ! ! & 7 & 7 ! 1 3 3 8 . 1 . % % 3 1 & 8 . 8 3 . . 3 !

1 ° . ° ! ! � � . & 3 % & 7 ! ° % % & & 8 7 8 ° ! & 3 � ° 3 . 1 %

! 8 . ! . & & 8 . 7 3 � ! 3 . 3 . & 1 & 3 & 8 & ! . 8 ! � 3 ! ° 1 3 . 1 % �

Page 280: Goldrei.classic Set Theory for Guided Independent Study

& ' & ' ) * + & ' , & & . * + , & . 2 ) ' . ) ) + * , ) + + 2 + ) , 2 & ) B

& ' + ) + . . 2 & + * ) B . 2 & + & . * & + 2

) , 2 , * * + + ' & ) . & + , 2 * � * )

, * ) & ' & ) . 2 & * 2 ) B * ) + + 2 , & + * & ) 2

* ' , ) + , & B 2 ) & ' , & & ' + B +

+ * ) + ) & ' + + * & . ) 2 * * ' ) ' + B . * + & ' + * *

+ ' , � � B + & B 2 & ) ) � + & +

2 ' + 2 + & * + ) 2

) + 2 & ' * + & . 2 � + & * � ) ) , & ) * ) + ) & ' + * ) + , ) & , B .

, � * . ' & + ) , & , � � + 2 & . � + + � + , & * B . 2 . � + *

+ ' + 2 + . , + 2 & ) . � � * & B & + * + + + 2 & & ) ' . 2 & ) B ) B . . 2 , �

& ' + ) ) , . 2 * 2 & . , & + � ) 2 + B + � ) * . ) ) ) +

. * & � + * ) ) & + B + � & ) ' + ) B +

� ' , ) B + + B . 2 2 . + * + ' + B + * , 2 '

' , &

+ & . 2 + * & . + & ' + ) 2 B * + ) ' . B + * � & . * , 2 . 2 2 . + + & ) ' . '

) B ) + & ' + 2 & ' + + & . ) 2 + + + 2 , 2 & + . 2 . . ) B 2 , �

2 + + ) + + + � ) + ) 2 * . . ) � ) , � . 2 2 . & +

* + & * & ' + 2 + + B . 2 2 . & + * + & + + � � ) B + + * ) ' , ' + + � � ) B + B . 2

. 2 + ) � * , & + 2 * ) B 2 . + + & ) ) , & + B . * + + � )

* . * + . , + 2 & & ) + + B ) + & + ) 2 + B * + ) ' + ) +

+ + B . 2 . & + * + & & ' + + . * , 2 ' , & + 2

' ) *

� 2 ' + + 2 & + * + 2 + + � � ) B + B + . & ) 2 + . 2 ) * +

) , 2 + . � + 2

+ & � + & * � ) & & + . ' ) & ) 2 . + . ' + ) B + * ) * & ) +

) 2 + 2 + ) + ' , � ' ) ' & & . . 2 . � + , & � � . + *

) & ' & ' + ' ) & ) B . 2 � + ' ) � . + ) , 2 * +

)

' + ' ) *

Page 281: Goldrei.classic Set Theory for Guided Independent Study

� � � � � " $ % � � % " � * + + * + + � % � * � $ % � % $ + $

* * ; � � % � � ; + $ $

" � $ + + � � % " $ + � %

� � � � $ % � % $ ; $ + + ; % %

" $ � * ; % " $ ; * + � � $ % % " $ ; $ +

� * � $ � " % " � % * � * $ *

* � � � " $ � * % ; % ; � $ % ; $ ; � % $ � ; % $ � %

; * + + % " " % ; $ % * ; � % * ; � $ �

� � " � * ; + + % " " * ; * $ $ + ; � � ;

� * � � % + �

% + � " % % % " % % + % � % %

; � % ; $ % % $ ; % + % +

% ;

� � + % % % � $ % ; + % % " $

; % + % * $ ; * * ; " $ ; + * " $ � $ $ � % " " $ �

% + � * � $ ; � % � + % % $ ; $

% ;

� " + + + % $ $ $

%

� ; %

� * + + � % � * � % $ ; $ + % ; % "

% + �

* * ; � � " ; $ " ; "

$ � ; % " $ $

� � " % + % %

� % � $ � $ % ;

% ; %

; * � � * �

� % � * � $ + " $

� � $ % � ; $ + % + $ $ � % + + % %

Page 282: Goldrei.classic Set Theory for Guided Independent Study

. / 1 2 3 1 / . . 1 / 7 9 7 1 < > 2 / 2 3 1 @ / 1 2 1 / 1 / . > 2 9 > @ / 7

/ > > / < / < 1 2 3 / @ 7 1 < @ 2 > 1 @ < / 1 / 9 3 9 < 9 7 7 9 2 1

/ 7 1 / @ 7

3 2 3 @ 2 3 / >

1 7 / / 9 7 2 3 9 / 9 7 2 3 @ 2 @ . 9 / . 9 7 9 > 1 < / . 9 > . 9 1

@ 1 @ > > 1 9 1 7 2 / / . 9 7 < 2 7 1 2 1 . 9 7 9 > 1 2 3 @ 2 / 7 9 7 7 9 1

1

3 9 / > > 7 . / 3 1 / 1 / 9 / 7 2 1 > 9 > <

1 3 @ 1 7 2 3 9 1 2 9 / 7 1 < 1 7 2 @ > > 7 9 1 1 . 1 @ 9 / 7 / 7 2 / . /

/ @ . 7 @ 7 2 1 9 @ . 9 2 3 9 @ 9 / 7 / @ 7 / /

. / > 1 < @ . 9 7 . 2 9 2 1 / 9 7 9 3 7 / 7 9

@ < 1 1 1 3 9 @ > 9 1 < 9 / 2 1 / 7 9 7 / 9 > >

/ 1 < > > / 3 1 . 2 9 < 2 / 9 2 9 /

/ 3 9 / /

< 9 7 / 2 2 / 7 2 / 7 / 1

< / 1 . 2 @ 7 < < 1 9 2 1 7 / 1 / 2 9 7 9

1 1 . 9 < 1 1 > < 7 2 1 7 9 > > /

1 9 1 7 / 7

1 1 @ 1 2 / @ 7 @ > 3 / 3 @ 9 < . 9 7 @ > 7 2 < < > < / @

< 1 / / . 9 / 9 @ 7 / 9 7 @ > 7 1 < 1 2 / 1 < . < 2 1 7

/ 7 2 @ 1 2 1 @ . 9 7 @ 7 < 1 / 7 3 1 7 9 2 1 < 7

1 2 1 7 < 1 7 > 1 1 2 1 / @ > 9 / 7 . / / 2 1 >

9 7 9 @ 1 . / 7 9 7 / > 1 7 2 /

@ 7 > 1 9 7 @ > > 1 @ 9 7 9 7 > / 1 7 9

< < 9 7 7 9 2 7 / 9 2 9 / 9 7 9

2 3 1 @ 7 1 . 9 < / @ . @ 7

/ 2 @ 2 3 1

3 1 1 > 2 / . 2 1 @ 7 < 2 3 @ 2 9 7 1 @ 7 1 7 1

/ 1 . 2 9 / 7 / 7 @ . 7 > <

@

/ . @ > @ 9 7 @ > < / @ < < 9 7 2

Page 283: Goldrei.classic Set Theory for Guided Independent Study

" 5 7 " " " > > " @ 5 A 5 A " @ " @

7 5 N " 7

T @ @ 7 5 7

5 A N e > A T @ > T f " 5 A A " A e >

7 7 @ 5 f f 5 @ 5 @ " e 7 T e

> @ T A 5 A " 5 A " A e > 5 A " A " T A e 5 A

7 5 T > A 5 " T

f T > 7 > @ A 5 e 5 A 5 5 T @ "

> @ @ > A @ 7 5 e T e > @ A 5 e 5 A 5 A e 5 A @ > " N T A " f " " A 5

T A " A 7 5 e 7 T > A 5 e " T f 5 N T A

" e " T @ " e e T A 5 N " T 7 @ "

e 7 " N " @ f e T 5 " e 5 " " T A 5 N @ e 7 " 5 N "

A 7 A @ " e 5 A 5 e 7 " 5 5 " @ 7 f f f 5 5 e 7 e @ N T e 7 " T 5 N

N 5 " 5 @ " T A e 5 N e A 5 e N " @ 5 f " N A 7 > @ f > " e > " 7 " 5 A e > A

N 5 f "

5 7 > @ T A T e 5 N 5 A @ " @ 5 e " N " T f A " T A 5 N T " e 5

@ " e e 7 T e T > A A > e " @ @ " e 5 e 7 " 5 A A > " > A " N 5 @ > e 7 " > e 7 " N

7 " T N T f " N 5 N > e @ " f

> > e > A A " 7 " A 5 N

f 5 7 " A e " @ @ " @ 5 e 7 T 7 " " 5 A " N " " f f > 5 5 " 5 e 7 " @ " e 5

T @ " @ A " " N @ > 5 A 5 7 > @ > @ e 7 T e e 7 " 5 " > e "

" " f f 5 " N " 5 N A @ T A " > " T @ @ " " A 5 " T T @

> 7 5 e 7 " @ " 5 " T f " A e " T f @ 7 " T N A T f 5 N

e 7 " N " e 5 " 5 f e " 5 @ 5 N e 5 > > " @ " e 5 T A e 5 5 A e >

7 5 7 @ > @ @ T f f T N " > T " T 7 " " 5 " @ @ 5 f f 5 @

A " 5 N 7 T 7 5 7 " @ > " e 7 e 7 " T N > T f " 5 A " A e T f " e 7 "

A " " @ > A f " N e T T T f 5 @ " @ e 7 T e 7 " @ T " T e e " N A

Page 284: Goldrei.classic Set Theory for Guided Independent Study

� � � � # % ' ) + + 2 3 � + % + 7 + ' 7 2 ) % 2 3 2 + #

� ' 2 3 3 + 7 � + ' 2 3 # + ) 2 7 +

� # 2 � ) % '

# � ) 7 � + # � ) 3 + 7 ' 3 ) 7 � # � 3 + ' � ) � + # + + % 7 +

) % 7 � + 2 � # ' � 3 + � 3 + ' % 2 7 � + 2 7 ) ) 2 % ' � # 7 � 7 �

7 # � + ) ' + � � + � + # 2 % + + 3 3 � + # + % � 7 2 # + � % ) %

' % 7 + # + 7 � 2 7 � + 2 7 ) 2 % ' ) % + + 7 + ' ' + + 7 ) % 7 # � 7 ) � % � +

� � � ) + % � + ) ' � + � 2 3 3 7 ' # 2 ) 2 7 ) � % ' 2 + 2 � # +

' % % 7 ' + � 2 % � + + ' � 3 7 ) � % � + ' + ) ' ' + ' ' + # 2 ' + + %

� # + � + 7 7 � % ) 3 + 7 � 2 2 % ) ) 2 7 +

' � % ' ) ' 7 + 7 � + % ' � ) ' ) � 2 %

' + 7 � 2 7 ' % ) � % ) ' + % � ) + 3 ' ) % � 3 + + % +

� + � # + ' � + ' ' + % 7 ) 3 2 % � 7 + # � + % + ) � + % � %

+ # � % 7 + % ) % � + + # ) 2 % 7 + 2 7 ) ) % 2 3

� � + % � � + 2 ) 2 ' ' % 7 � 2 7 ) � % ' + % 7 7 2 2 % 2 %

7 � ' 2 % % � 7 + # � % + + 2 % % � + # � + # �

+ + � % + 2 ' % + 7 2 �

� % ) ' + 7 � + % ' � + ) � � + % + 2 � % � 2 % 2 3 ' �

3 � ) � � + % + � % �

� + % ' � # � ) + 2 + 7 � � � + 2 3 3 + ) � + � 2 3 % � 2 7 7 + 7

7 � + # + % 7 � ) ' � � � ) ' + 7 � � + + % + # ) 7 3 ) % ' � � ) % 7 � +

% + + + % + � 2 � ' ' + 7 + � # + ) # ) % ) 3 + ' # � ) � ' ' 7 + 3 +

+ ' � 2 3 � + + # ) + 2 # ) + ) + 2 � � � + 3 # � + � + � % ' ) ' + % �

� 7 � + � ' � + + ) � ) � + � % 7 # 7 ) � % +

2 7 ) + � ) + # # � � + 7 ' 7 + � + ) � % 7 � # + 2 7 ) 2 3 3 +

7 � + 7 � � + 7 2 + ' � 7 + � + # 2 # � # ) 7 7 + % 2 '

2 � % 2

% + � # + % � + 7 + � ) + # � ) ' � 2 ) 7 ' � � ' � � ) + 7 +

# ' � � % ' # 2 3 3 � ' ) 3 + + 3 ) % # � � + + ' +

' % � + ' 7 ' % � % + 3 ' � 7 # 7 ) 3 + � + # 2 # � � + ' 7 + ' +

� # 7 � 7 � ) � ) % � # + 3 ) + 3 2 ' ' � ' + ) 7 � 2 � 7 ) 7 + # ) + 2

� � 2 % ' 7 7 + ' + 7 ' 7 3 + + 3 # � 7 � + ' + 7 ' 2 7 3 � + # 3 + + 3 ' + 3 '

� % ' # 7 ) � � + ' � + ' 7 + + ' � 2 7 � 7 � + ' 7 7 � + % 3 ) ) 7

� % ' 2 + ' ' 2 7 � + ' + ' ' � # � # ) % 2 3 ' 7 2 + + # + 2 2 3 � + + 2 7

+ ' � % 7 � 2 7 � + � % ' 7 # ) � % ' ) +

Page 285: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � % � � � � � � � � � � � � � � � �

� 2 % � � � 2 � � � � � � � � � � � % � � � � � 2 � � � 2 � � � � � � � �

� � � � � � % � � % � � � � � % � 2 � 2 � � � � � 2 � � � % � � � � %

� � � � � � � � � � � 2 � � � 2 % � � � � � % 2 � � � � � � � 2 % 2 � �

� � 2 � � � � � � � � 2 � � % % � � � � � 2 � % 2 � � � � 2 2 % �

2 � � % � � � � � � � � 2 � � � � � � � � � % 2 � % � � �

2 � � � � � � � � � 2 � 2 � � � � � � � � � � � � � � � � � � � � � � �

� � � � � � � % � � � % � � � � � � � � � � � � � 2 � � % �

2 � � � � � � 2 � � � � � � � � � � � � � � � � � 2 � � � 2 � 2 % � �

2 � 2 � � � 2 � � � � � � � � � % � � � � � � 2 � � � � 2 � � � � � � � � � � � �

� 2 % � � � � � � 2 % % � � � � � � � � 2 � � � 2 � � � �

% � � 2 2 � � 2 % � � � � � � � % 2 � � � % � � � � � � �

� � � � � % % � � � % � � � � � � � � � � � � � 2 � 2

� � � � � 2 � � � � � � � � � � � � � � � � % 2 � �

� � � � � � 2 % � � � � � � � � � � � � � � � % � � � � � � � � � � �

� � � � � � � � � � � � � � � � 2 � � � � � � � � � %

� � � � � � � � � 2 � � � � � � � % � � � 2 � 2 � � � 2 � 2 % �

� � � � 2 � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � � � � � % � � � � � � � � � � � � � � � � �

� � � � � � � � � 2 � � � � � � � �

2 � � � � � � � � � � � � � � � � � �

� � � � � � � � � � � � � % � � � � �

� � � � � � � � � � � � � � � % % � �

� � � � � 2 � � � � 2 % � � � �

� % � � � � � �

� % 2 � � � � � � 2 � � � � � � � 2 �

� � �

� � 2 % � � � � 2 %

� � � � 2 � � % � � � � � � � � � � % 2 �

% % � � � � % � � � � � 2 % % � � � � 2 � � � � � �

2 �

� � � � � � � � � � 2 � � � � � � � � � � � � � � � 2 � �

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� � � 2 � � � � � � � � � � 2 � � � � � 2 � � 2 � 2 � � 2 % �

2 � � � � 2 % % � � � � � � � � � 2 � � � � % � � 2 %

Page 286: Goldrei.classic Set Theory for Guided Independent Study

� � � � � � � � � � � � � � � " � � $ & $ � � & � $ & � $ � �

" � � � $ " � � & & � � � � � � � � $ � � � $ � � � & � � � � �

$ " � " $ � � � � � � � � � � � � � � � � $ � � � � � � � � � �

& � � � � � " � � $ � & $ � � & � $ " � � � � � $

" � �

� $ " & � & � � � � � � & � � $ & � $ $ � � � $ " &

& � " � $ � $ � � " � � " � & � � � " � & � � � � � � $ � $

� � $ � & � � � � � $ & & � � � � � $ & � " � �

� � � � � � & $ � � � � � � � & $ � � � �

" � & � � � $ $ " � � � � � � � � & � � & � � � � � � $ � � $ " &

� $ � $ � � � � � $ � � � � & � � � � � � � & &

� � � � � � & � � � � � � � � � � $ � $ � � $ � & �

$ & � $ � � � & � � & $ � � � � � � � $ � � � � � � � � �

� � � � � " � � $ $ & & � � � � � � " � � � $ � � $ & � �

� � � � � � " & � � � � � $ " � � � " � � � � � & � � � & $ � �

� � $ � $ " � � " � " � � � $ � � � $ " �

" � � � � � $ � $ � $ � � � � � � � $ � � � � � � � $ � � � �

� � � � � & & $ " � � � $ & $ $ " � � � � � " � � � � � � � $ � $ �

$ " � � � � � $ " � � & � � � � � & & $ " � " � � "

� � $ � � � & � $ � & $ " � " � � � � & $ " � � �

$ � � � � & $ � � $ " & & � � � $ " � � $ $ �

� � � � & � � � � $ � � � � � " � � � $ � � � � �

� � & & $ " � " � � � � � $ � & � � & $ $ & �

& � � � & " $ " $ � & & $ " � " � $ � �

� � & � � � $ � � � � & & $ " � " $ � � � � � � � $ � & &

" � " � & & $ " � " � $ " & � � � $ " � � & � � � �

$ " � � � & & $ " � " � � & $ " � � $ " � � � � � � $ "

� � � � � � $ � $ � � � � � & & $ " � " " � & � � $ � $ �

" � � � � � $ � " � � � � � � � � � � � $ � $ � � � " � & � � $ � � � � &

� $ � � � � $ � � & � & & � $ "

� � � � � $ � � � � � � � & & $ � " & $ �

� � " � � & � � � � � � � � � � � $ $ � & $ & $ " " � � " &

� � � $ � � � $ � � " $ $ $ � " � $ & & $ � �

� � � � � � " � � $ � � � � & & � � $ $ � �

� � $ � � � � � � � $ � & $ � � � & $ $ � �

$ & & $ � � " � � � � � � � � � " � & � � � � $ � � & $ " �

� $ $ $ � � � & � $ � " � � � � � � � � � � � � " � $ $ � � � � � � � �

� � $ � � � � � � � � $ � � � � $ � � � � � � " � $

" � � " $ " � � � � � � " � � � & & $ � " � & � � $ � " � � �

$ " $ � � � � � $ � � � � � � � � $ � �

"

� � $ " � � $ � & $ " � $ � � � & � $ � � � � � �

� � $ � � & � � � � � & � � � � � �

Page 287: Goldrei.classic Set Theory for Guided Independent Study

� � � $ ( $ � + $ / � � $ $ � + �

+ $ � / � / + � + G G + � / N $ / � + N � � + $

$ ( $ / + � + $ + W G / $ $ � $ ( G � G $

$ / � + / G � _ G _ + � � N / $ N _ �

$ N $ + $ � � / � + N G / � $ N _ + $ $ $ G

� $ + $ N _ + N + + $ G � G � + G + N � N � $ +

$ $ $ / + $ � � + $ ( $ � _ / $ $ $ ( $ � ( $ _ ( $

$ $ $ � + $ G G _ / � G � $ � � N � + $ � � G _ � /

+ � + $ + � W $ / $ ( $ � G $ � � $ G � � G � ( G $ + $ � ( + G

( � � _ � $ + $ _ G $

/ + $ � � $ / � + $ $ $ � $ G ( $ + ( $ � W � G

G � / G / � + $ N _ G � +

� � + $ � $ N _ � � � N G G $ � + _ � + N _ $ G � $ G G $ $ (

� � + $ ( $ � $ � ( N N + $ N W ( � $ + � / / � $ ( $ G � W $ �

� $ $ G $ G _ N / / � ( $ � W $ $ $ ( � W $ � $

� $ � ( / � � W $ N � N G $ � � + $ _ _ � N � G � $ W G _ / $ $ / $ G G � + $

� ( $ G / � � + $ + / $ G � � $ / $ � $ � ( N _ $ / �

N � + $ N W $ $ + G � $ G � $ G � N � $ / N � � �

_ $ $ � $ N � $ G � � $ � + $ � � $ W

/ $ � + $ _ G + _ G � � � � $ � / � ( � N � $ ( / $ ( $

$ � $ $ / G N / ( G � $ � ( � + $ N $ � $ N $ _ N

� $ _ / / � + $ / � � � + � ( � / / _ $ $ G + $

$ � _ / G + + G � N ( � G $ N � � + $ $ + � ( � + $ + $ + $ ( $ N

$ ( $ $ W � $ $ N / � � $ ( _ / / + $ � G _ $ G _ $ � _ / / � $

G G $ � � / $ � � � $ $ N W � � $ G � _ � / � + / $ $ $

_ N $ / � G $ G $ N � + $ � _ N / G + � W / � N W $ � � + � + �

+ N � G W $ G $ � + $ � ( / $ + � $ � + N � � + � $ _ / � $ ( $ $ _ / / /

� / _ / $ $ $ W / G � $ � ( $

Page 288: Goldrei.classic Set Theory for Guided Independent Study

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Page 289: Goldrei.classic Set Theory for Guided Independent Study

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Page 290: Goldrei.classic Set Theory for Guided Independent Study

Z ^ a d Z f g Z Z

r

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Page 291: Goldrei.classic Set Theory for Guided Independent Study

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Page 292: Goldrei.classic Set Theory for Guided Independent Study

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Page 293: Goldrei.classic Set Theory for Guided Independent Study

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Page 294: Goldrei.classic Set Theory for Guided Independent Study

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Page 295: Goldrei.classic Set Theory for Guided Independent Study
Page 296: Goldrei.classic Set Theory for Guided Independent Study

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