set theory chapter 2. day 1 set – collection school of fish gaggle of geese pride of lions pod of...

SET THEORY

Chapter 2

DAY 1

Set – collection

• School of fish

• Gaggle of geese

• Pride of lions

• Pod of whales

• Herd of elephants

• Set – usually named with a capital letter.

• Well defined

A is the set of the first three lower case letters of the English alphabet.

• Elements of the set

A is the set of the first three lower case letters of the English alphabet.

a, b, and c are elements of set A

Ac

Ab

Aa

• Natural Numbers (Counting Numbers)

N = {1, 2, 3, . . . }

Three ways of defining a set:

• List

A = {1,2,3}

• Description

A is the set of the first three counting numbers.

• Set Builder Notation

}4,|{ xNxx

• Universe

• Empty set

Example

The set of natural numbers greater than 12 and less than 17.

Example

{x | x = 2n and n = 1, 2, 3, 4, 5}

Example

{3, 6, 9, 12, . . . }

Example

The set of the first 10 odd natural numbers.

Venn Diagrams

C

BA

BA

A

Set A

A

Complement of A

}|{ AxxA

A

A intersect B

}|{ BxandAxxBA

BA

A Union B

BA

}|{ BxorAxxBA

Disjoint Sets

BA

BA

Subsets

A is a subset of B if every element of A is also an element of B.

BA

List all the subsets of {a,b,c}

• { }• {a}• {b}• {c}• {a,b}• {a,c}• {b,c}• {a,b,c}

List all the subsets of {a,b,c}

• Proper Subsets:

{ } , {a} , {b} , {c} , {a,b} , {a,c} , {b,c}

• THE Improper Subset:

{a,b,c}

Subset Notation

Let A = {a,b,c}

{a} A“The set of a is a subset of A.”(think: The set of a is a proper

subset OR IS EQUAL TO A.)

{a} A“The set of a is a proper subset of A.”

True or False? A = {b,c,f,g}

{b,f} A

{b,f} A

True or False? A = {b,c,f,g}

{b,f} A True

{b,f} A True

True or False? A = {b,c,f,g}

{b,d} A

True or False? A = {b,c,f,g}

{b,d} A False

Because d A

True or False? A = {b,c,f,g}

{b,c,f,g} A

True or False? A = {b,c,f,g}

{b,c,f,g} A True

{b,c,f,g} A

True or False? A = {b,c,f,g}

{b,c,f,g} A True

{b,c,f,g} A False

Because {b,c,f,g} = A

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

C

BA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

CA

CA

BA

BA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

?)( CBA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

CB

CBA )(

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

)(

},{

CBA

urCB

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

?BA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

BA

BA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

BA

rqBA },{

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

?

},,,,,,,,{

BA

zyxwvutspBA

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

B

A

BA ?

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

BA

zyxwvpB

zyxwvutsA

},,,,,{

},,,,,,,{

U = {p,q,r,s,t,u,v,w,x,y,z}A = {p,q,r}, B = {q,r,s,t,u}, C = {r,u,w,y}

BABA

zyxwvBA

zyxwvutspBA

},,,,{

},,,,,,,,{

C

BA

)( CBA

C

BA

)()( CABA

C

BA

)( CBA

C

BA

CBA )(

DAY 2

Homework QuestionsPage 83

Three types of numbers.

• Nominal

• Ordinal

• Cardinal

“The student with ticket 50768-973 has just won second prize – four tickets to the big

game this Saturday.”

Three types of numbers.

• Nominal – name or label – for identification• Ordinal – tells what order it comes in

relation to the rest.• Cardinal – Answers the question “how

many?”

“The student with ticket 50768-973 has just won second prize – four tickets to the big

game this Saturday.”

Cardinality of the Set

If a cardinal number answers the question “how many?” then the cardinality of a set will tell us how many elements are in the set.

The notation for “the cardinality of set A” (or the number of elements in A) is

n(A)

Equal Sets

Two sets are equal if the have the exact same elements.

Example:

A = {a,b,c} and B = {c,a,b}

then A = B

Consider A = {a,b,c} and C = {x,y,z}

They are not equal because they do not have the same exact elements.

What characteristic do they share?

Equivalent Sets

A and C have the same number of elements. Their cardinality is the same.

n(A) = 3 and n(C) = 3

n(A) = n(C)

A and C are equivalent sets.

CA

CA

~

If two sets are equivalent, you can set up a one-to-one correspondence between them. (That is, you can match them up in pairs.)

z

y

x

c

b

a

There are actually 6 different one-to-one correspondences you can set up between these two sets. (6 ways that you can make pairs.)

A = {a,b,c} and C = {x,y,z}

(make an orderly list)

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}

a – x

b – y

c – z

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}

a – x a - x

b – y b - z

c – z c - y

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}

a – x a – x a - y

b – y b – z b - x

c – z c – y c - z

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}

a – x a – x a – y a - y

b – y b – z b – x b - z

c – z c – y c – z c - x

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}a – x a – x a – y a – yb – y b – z b – x b - zc – z c – y c – z c – x

a – zb – xc – y

6 different one-to-one correspondences:

A = {a,b,c} and C = {x,y,z}a – x a – x a – y a – yb – y b – z b – x b - zc – z c – y c – z c – x

a – z a - zb – x b - yc – y c - x

A = {x|x is a moon of Mars}

B = {x|x is a former U.S. president whose last name is Adams}

C = {x|x is one of the Bronte sisters of nineteenth-century literary fame}

D = {x|x is a satellite of the fourth-closest planet to the sun}

Which of these sets are equal and which are equivalent?

What do we need to know about each set to answer this question?

A = {x|x is a moon of Mars}

A = {Deimos, Phobos}

n(A) =

A = {Deimos, Phobos}

n(A) = 2

B = {x|x is a former U.S. president whose last name is Adams}

B = {John Adams, John Quincy Adams}

n(B) =

A = {Deimos, Phobos}

n(A) = 2

B = {John Adams, John Quincy Adams}

n(B) = 2

C = {x|x is one of the Bronte sisters of nineteenth-century literary fame}

C = {Anne, Charlotte, Emily}

n(C) =

A = {Deimos, Phobos}

n(A) = 2

B = {John Adams, John Quincy Adams}

n(B) = 2

C = {Anne, Charlotte, Emily}

n(C) = 3

D = {x|x is a satellite of the fourth-closest planet to the sun}

D = {Deimos, Phobos}

n(D) =

A = {Deimos, Phobos}

n(A) = 2

B = {John Adams, John Quincy Adams}

n(B) = 2

C = {Anne, Charlotte, Emily}

n(C) = 3

D = {Deimos, Phobos}

n(D) =2

Finite/Infinite

• Whole numbers?

• Real numbers between 0 and 1?

• Factors of 20?

• Multiples of 20?

• Number of grains of sand on the earth?

Example 2.9Page 94

• n(U) = 60• n(S) = 24• n(E) = 22• n(H) = 17• 5 both S and E• 4 both S and H• 3 both E and H• 2 all three

H

ES

Attribute Lab

Three attributes considered are• Size• Color• Shape

HexagonYellow

Yellow andHexagon

HexagonYellow

Yellow or Hexagon

HexagonYellow

• A and B – You must get through the first door AND the second door. (more restrictive)

• A or B – You may go in the first door OR the second door. (more generous)

RECTANGLEBLUE

Day 3

Homework Questions Page 97

Binary Operations

• Addition

• Subtraction

• Multiplication

• Division

__________ + __________ = __________

Addend + Addend = Sum

__________ - __________ = __________

Addend + Addend = Sum

Minuend – Subtrahend = Difference

__________ X __________ = __________

Addend + Addend = Sum

Minuend – Subtrahend = Difference

Factor X Factor = Product

__________ __________ = __________

Addend + Addend = Sum

Minuend – Subtrahend = Difference

Factor X Factor = Product

Dividend Divisor = Quotient

PropertiesPages 104 and 120

• Closure

Counting Numbers = {1, 2, 3, . . . }

Whole Numbers = {0, 1, 2, 3, . . . }

Closure Examples

Is the set of Whole Numbers closed with respect to

• Addition?

• Subtraction?

• Multiplication?

• Division?

Closure Examples

Is the set of Even Counting Numbers closed with respect to

• Addition?

• Subtraction?

• Multiplication?

• Division?

Closure Examples

Is {0, 1} closed with respect to

• Addition?

• Subtraction?

• Multiplication?

• Division?

PropertiesPages 104 and 120

• Closure• Commutative• Associative• Identity Element for Addition• Identity Element for Multiplication• Multiplication-by-Zero Property • Distributive Property of Multiplication over

Addition

Examples

2 + (3 + 4) = 5 + 4

Examples

2 + (3 + 4) = 5 + 4 Associative

2 + (3 + 4) = 7 + 2

Examples

2 + (3 + 4) = 5 + 4 Associative

2 + (3 + 4) = 7 + 2 Commutative

2(3 + 4) = 6 + 8

Examples

2 + (3 + 4) = 5 + 4 Associative

2 + (3 + 4) = 7 + 2 Commutative

2(3 + 4) = 6 + 8 Distributive

2(3 + 4) = (7)2

Examples

2 + (3 + 4) = 5 + 4 Associative

2 + (3 + 4) = 7 + 2 Commutative

2(3 + 4) = 6 + 8 Distributive

2(3 + 4) = (7)2 Commutative

Conceptual Models

• Addition– Set Model

Conceptual Models

• Addition

• Subtraction (page 108)– Take-away– Missing Addend– Comparison– Number-line

Take-away - Missing AddendComparison - Number-line

Identify which model would illustrate the problem best.

Mary got 43 pieces of candy. Karen got 36 pieces. How many more pieces does Mary have than Karen?

Take-away - Missing AddendComparison - Number-line

Identify which model would illustrate the problem best.

Mary gave 20 pieces of her 43 pieces of candy to her brother. How many pieces does she have left?

Take-away - Missing AddendComparison - Number-line

Identify which model would illustrate the problem best.

Karen’s older brother collected 53 pieces. How many more pieces would Karen need to have as many as her brother?

Take-away - Missing AddendComparison - Number-line

Identify which model would illustrate the problem best.

Ken left home and walked 10 blocks east. The last 4 blocks were after crossing Main Street. How far is Main Street from Ken’s house?

Conceptual Models

• Addition

• Subtraction

• Multiplication (page 115)– Repeated Addition– Number-line– Rectangular Array– Multiplication Tree

Multiplication TreeMelissa has 4 flags colored red, yellow, green and blue. How many ways can she display them on a flagpole?

Blue

Green

Yellow

Red

Blue-Green

Blue-Yellow

Blue-Red

Green-Blue

Green-Yellow

Green-Red

Yellow-Blue

Yellow-Green

Yellow-Red

Red-Blue

Red-Green

Red-Yellow

Conceptual Models

• Addition

• Subtraction

• Multiplication– Repeated Addition– Number-line– Rectangular Array– Multiplication Tree– Cartesian Product

Cartesian Product

• The Cartesian Product of A and B is a set of ordered pairs written A X B, and read “A cross B.”

• A X B = {(a,b) | a A and b B}

Cartesian Product

• A X B = {(a,b) | a A and b B}

Example:

A = {5, 6, 7} B = {6, 8}

A X B = {(

Cartesian Product

• A X B = {(a,b) | a A and b B}

Example:

A = {5, 6, 7} B = {6, 8}

A X B = {(5,6), (5,8), (6,6), (6,8), (7,6), (7,8)}

Cartesian Product

Example:

A = {5, 6, 7} B = {6, 8}

A X B = {(5,6), (5,8), (6,6), (6,8), (7,6), (7,8)}

NOTE:

n(A) = 3 , n(B) = 2 and n(AXB) = 6

How many different things can you order at the yogurt shop if you must choose from a waffle cone or a sugar cone and either vanilla, chocolate, mint, or raspberry yogurt?

C = {w, s}, Y = {v, c, m, r}

Cartesian ProductC = {w, s}, Y = {v, c, m, r}

C X Y = {(w, v), (w, c), (w, m), (w, r), (s, v), (s, c), (s, m), (s, r)}

n(C X Y) = 8

Conceptual Models

• Addition

• Subtraction

• Multiplication

• Division (Page 121)– Repeated Subtraction– Sharing– Missing Factor

Division Example

• Describe how you would divide 78 by 13 using counters and each of the following models.

– Repeated Subtraction– Sharing– Missing Factor

Family of Facts

20 4 = 5 5 X 4 = 20

and

20 5 = 4 4 X 5 = 20

Family of Facts

0 ÷ 4 = 0 and 0 X 4 = 0

4 ÷ 0 = ??

and ?? X 0 = 4

Division by Zero is Undefined.

Extra Practice Worksheet

DAY 4

HomeworkPages 111 and 130

Worksheet Answers

4.3.

2.1.

A

B C CB

A

A

B CCB

A

8.7.

6.5.

A

B C CB

A

A

B CCB

A

Math and MusicThe Magical Connection!

• Scholastic Parent and Child Magazine

• Spelling

• Phone Numbers

• School House Rock

“Skip to My Lou”

Chorus: Times facts, they’re a breeze;

Learn a few, then work on speed.

Times facts, you’ll be surprised

By just how fast you can memorize.

3 time 7 is 21Now, at last we’ve all begun.4 times 7 is 28Let’s sing what we appreciate.

(Chorus)

5 times 7 is 35.Yes, by gosh, we’re still alive.6 times 7 is 42.I forgot what we’re supposed to do.

(Chorus)

Print Review for Test

Venn Diagram Lab