definition:sets a set is a well-defined collection of objects. examples: 1.the set of students in a...
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DEFINITIONDEFINITION::SETSSETS
A SET IS A WELL-DEFINED COLLECTION OF OBJECTS.
EXAMPLES:
1. THE SET OF STUDENTS IN A CLASS.
2. THE SET OF VOWELS IN ENGLISH ALPHABETS.
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REPRESENTATION OF SETSREPRESENTATION OF SETS
TYPES OFREPRESENTATION
OF SETS
TABULAROR
ROSTER FORM
SET-BUILDER OR
RULE METHOD
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TABULAR OR ROSTER FORMTABULAR OR ROSTER FORM
IN THIS METHOD,WE LIST ALL THE ELEMENTS OF THE SET SEPERATING THEM BY MEANS OF COMMAS AND ENCLOSING THEM IN CURLY BRACKETS { }.
EXAMPLE:-
IF A IS THE SET CONSISTING OF THE PRIME NUMBERS BETWEEN 1 AND 10,THEN THE SET A CAN BE WRITTEN IN TABULAR FORM
AS A={2,3,5,7}.
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SET-BUILDER OR RULE SET-BUILDER OR RULE METHODMETHOD
IN THIS METHOD,INSTEAD OF LISTING ALL ELEMENTS OF A SET,WE WRITE THE SET BY SOME SPECIAL PROPERTYOR PROPERTIES SATISFIED BY ALL ITS ELEMENTS AND WRITE IT AS
A={x: P(x)}
A={x |x has the property P(x)}
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TYPES OF SETSTYPES
OF SETS
FINITE SET
FINITE SET
SINGLETONSET
SINGLETONSET
EMPTYSET
EMPTYSET
INFINITESET
INFINITESET
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1.FINITE SET-IF THE ELEMENTS OF A SET ARE FINITE IN NUMBER,THEN THE SET IS CALLED A FINITE SET.
EXAMPLE:-
{1,5,25,125} IS A FINITE SET.
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2. INFINITE SET-IF THE ELEMENTS OF A SET ARE INFINITE IN NUMBER,THEN THE SET IS CALLED AN INFINITE SET.
EXAMPLE:-
SET OF NATURAL NUMBERS, N={1,2,3,...}
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3.SINGLETON SET-A SET CONSISTING OF ONLY ONE ELEMENTIS CALLED A SINGLETON SET.
EXAMPLE:-A={2} IS A SINGLETON SET.
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4.EMPTY SET-A SET CONSISTING OF NO ELEMENT
IS CALLED AN EMPTY SET AND IS DENOTED AS Φ
OR { }.
EXAMPLE:-
THE SET OF ALL ODD INTEGERS GREATER THAN 7
AND LESS THAN 9 IS AN EMPTY SET.
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EQUAL SETSEQUAL SETS
TWO SETS A AND B ARE SAID TO BE EQUAL ,IF
EVERY ELEMENT OF A IS AN ELEMENT OF B
AND EVERY ELEMENT OF B IS AN ELEMENT OF
A.EXAMPLE:
{3,7,9}={7,9,3}
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CARDINAL NUMBER OF A FINITE SETCARDINAL NUMBER OF A FINITE SET
THE NUMBER OF ELEMENTS IN A FINITE SET A IS KNOWN AS
CARDINAL NUMBER OR ORDER OF A FINITE SET AND IS
DENOTED BY n(A).
EXAMPLE :
IF A={1,2,3,4} THEN n(A)=4
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EQUIVALENT SETSEQUIVALENT SETS
TWO FINITE SETS A AND B ARE SAID TO BE EQUIVALENT SETS IF THE NUMBER OF ELEMENTS IN A IS EQUAL TO THE OF ELEMENTS IN B i.e.,n(A)=n(B) AND EQUIVALENCE IS DENOTED BY ~.
EXAMPLE :
IF A={1,2,3} AND B={X,Y,Z},THEN n(A)=n(B)=3
SO,A~B.
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SUBSETSSUBSETS
THE SET B IS SAID TO BE THE SUBSET
OF A IF EVERY ELEMELEMENT OF SET B IS ALSO AN ELEMENT OF A AND WE WRITE IT AS AB OR BA .IF B IS NOT A SUBSET OF A,THEN WE WRITE B⊈A.
EXAMPLE:
IF A={1,2,3,4,5} AND B={1,2,3} THEN BA.
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PROPER SUBSETPROPER SUBSET
A SET B IS SAID TO BE A PROPER SUBSET OF SET A,IF EVERY ELEMENT OF SET B IS AN ELEMENT OF A WHEREAS EVERY ELEMENT OF A IS NOT AN ELEMENT OF B.
EXAMPLE:
{2}{2,3,4}
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POWER SETPOWER SET
THE COLLECTION OF ALL SUBSETS OF A SET A IS CALLED THE POWER SET OF A.IT IS DENOTED BY P(A).
EXAMPLE:
IF A={1,2,3}, THEN
P(A)={Φ,{1},{2},{3},{1,2},{2,3},{1,3},{1,2,3}}
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THEOREMTHEOREM:: PROVE THAT THERE ARE 2PROVE THAT THERE ARE 2n n ELEMENTS IN ELEMENTS IN THE CLASS OF ALL SUBSETS OF A SET OF n THE CLASS OF ALL SUBSETS OF A SET OF n ELEMENTS.ELEMENTS.
PROOFPROOF:CONSIDER A SINGLETON SET A={a}.IT :CONSIDER A SINGLETON SET A={a}.IT HAS TWO POSSIBLE SUBSETS HAS TWO POSSIBLE SUBSETS ΦΦ AND {a}. AND {a}.LET CLASS OF ALL SUBSETS OF SET A BE LET CLASS OF ALL SUBSETS OF SET A BE DENOTED AS P(A).DENOTED AS P(A).THUS, P(A)={THUS, P(A)={ΦΦ,{a}},{a}}IF A HAS ONE ELEMENT ,THEN P(A) HAS 2 IF A HAS ONE ELEMENT ,THEN P(A) HAS 2 ELEMENTS.ELEMENTS.CONSIDER SET A={a,b}.CONSIDER SET A={a,b}.IT HAS 4 POSSIBLE SUBSETS IT HAS 4 POSSIBLE SUBSETS ΦΦ ,{a},{b},{a,b} ,{a},{b},{a,b}
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P(A)={Φ ,{a},{b},{a,b}} IF A HAS 2 ELEMENTS, THEN P(A) HAS 22 ELEMENTS.SIMILARLY,IF A={a,b,c},THEN P(A)={Φ,{a},{b},{c},{a,b},{b,c},{c,a},{a,b,c}} IF A HAS 3 ELEMENTS ,THEN P(A) HAS 23 ELEMENTS.PROCEEDING THIS WAY WE PROVE THAT,IF A HAS n ELEMENTS THEN P(A) HAS 2n ELEMENTS.
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COMPARABLE SETSCOMPARABLE SETS
TWO SETS A AND B ARE SAID TO BE COMPARABLE IF ONE OF THEM IS SUBSET OF THE OTHER i.e.,EITHER AB OR BA.
EXAMPLE:
THE SETS {1,3,4,5} AND {1,2,3,4,5,6} ARE COMPARABLE SETS.
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DISJOINT SETSDISJOINT SETS
IF A AND B ARE TWO SETS SUCH THAT THERE ARE NO COMMON ELEMENTS IN A AND B,THEN THESE ARE CALLED DISJOINT SETS.
EXAMPLE:
A={a,b,c,d} AND B={e,f,g,h}.
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UNIVERSAL SETUNIVERSAL SETWHEN ALL THE SETS UNDER
CONSIDERATION ARE SUBSETS OF A LARGER SET THEN THIS LARGER SET IS CALLED THE UNIVERSAL SET.IT IS DENOTED BY U.
EXAMPLE:LET
U={1,2,3,4,5,6,7,8},A={1,2,3},B={4,5,6},C={7,8}HERE A,B,C ARE SUBSETS OF U,THEN U IS
THE UNIVERSAL SET.
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COMPLIMENT OF A SETCOMPLIMENT OF A SET
COMPLIMENT OF A SET A IS THE COLLECTION OF ELEMENTS OF U WHICH ARE NOT IN A.IT IS DENOTED BY A׀
A׀={x:xU,xA}
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EXAMPLESEXAMPLES::
EX.1.WRITE THE FOLLOWING SETS IN ROSTER FORM:
a) A={x:x IS AN INTEGER AND -3<x<7}.
b) B={x:x IS A MULTIPLE OF -5 AND |x|20}.
SOL.:a)the integers between -3 and 7 are
-2,-1,0,1,2,3,4,5,6
ROSTER FORM OF SET A={-2,-1,0,1,2,3,4,5,6}.
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b)b)|x||x|20 20 -20 -20 xx2020ALSO,x IS A MULTIPLE OF -5.ALSO,x IS A MULTIPLE OF -5.
B=SET OF ALL MULTIPLES OF -5 WHICH LIES B=SET OF ALL MULTIPLES OF -5 WHICH LIES BETWEEN -20 AND 20BETWEEN -20 AND 20 ROSTER FORM OF SET B={-20,-15,-10,-ROSTER FORM OF SET B={-20,-15,-10,-5,0,5,10,15,20}5,0,5,10,15,20}
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• EX-2.WRITE THE FOLLOWING SETS IN SET BUILDER FORM:
a) {5,10,15,20} b){14,21,28,35,42…,98}
SOL.a)LET A={5,10,15,20}
Now,5=51,10=52,15=53,20=54
A={x:x=5n,n4,nN}
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b)LET B={14,21,28,35,42,…,98}
WE OBSERVE THAT ALL THE ELEMENTS OF SET B ARE NATURAL NUMBERS,MULTIPLES OF 7 AND LESS THAN 100.
B={x:x IS A MULTIPLE OF 7 AND 7<x<100,xN}.
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EX.3:WRITE DOWN ALL THE SUBSETS OF {1,2,3}.
SOL.: ALL POSSIBLE SUBSETS OF {1,2,3} ARE:Φ ,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}.
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EX.4: IF U={1,2,3,4,5,6,7},FIND THE COMPLIMENT OF FOLLOWING SETS:
a)A={1,2,3} b) B={6,7}
SOL.: a)HERE U={1,2,3,4,5,6,7} AND A={1,2,3}A }4,5,6,7={׀
b) HERE B={6,7}
B }1,2,3,4,5={׀
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VENN DIAGRAMVENN DIAGRAM
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VENN DIAGRAM OF AU
A
U
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VENN DIAGRAM OF AVENN DIAGRAM OF ABB
A
B
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UNION OF SETSUNION OF SETS
LET A AND B BE TWO GIVEN SETS.THEN THE UNION OF A AND B IS THE SET OF ALL THOSE ELEMENTSWHICH BELONG TO EITHER A OR B OR BOTH.
AB={x: EITHER xA OR xB}
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A B
U
AB
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INTERSECTION OF SETSINTERSECTION OF SETS
LET A AND B BE TWO GIVEN SETS.THEN INTERSECTION OF A AND B IS THE SET OF ELEMENTS WHICH BELONG TO BOTH A AND B.
AB={x:xA AND xB}
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AB
AB
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APPLICATION OF SETSAPPLICATION OF SETS
• IF A AND B ARE NOT DISJOINT SETS THEN
n(AB)=n(A)+n(B)-n(AB)
• IF A AND B ARE DISJOINT SETS THEN
n(AB)=n(A)+n(B).
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EXAMPLEEXAMPLE
• IN A GROUP OF 65 PEOPLE,40 LIKE CRICKET, 10 LIKE BOTH CRICKET AND TENNIS.
a) HOW MANY LIKE TENNIS?
b) HOW MANY LIKE TENNIS ONLY AND NOT CRICKET?
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SOL.:LET A BE THE SET OF PEOPLE WHO LIKE CRICKET AND B BE THE SET OF PEOPLE WHO LIKE TENNIS.
THEN,
n(AB)=65
n(A)=40
n(AB)=10
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a) WE KNOW THAT, n(AB)=n(A)+n(B)-n(AB) 65 =40+ n(B)-10 n(B)=35HENCE,35 PEOPLE LIKE TENNIS.b)NUMBER OF PEOPLE WHO LIKE ONLY
TENNIS=n(B)-n(AB)=35-10=25HENCE,NUMBER OF PEOPLE WHO LIKE TENNIS ONLY AND NOT CRICKET IS 25.
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ASSIGNMENTASSIGNMENT
• DEFINE SETS WITH EXAMPLES?• WHAT IS THE DIFFERENCE BETWEEN
PROPER SUBSET AND IMPROPER SUBSET ?
• IF A={1,2,3} THEN FIND THE POWER SET OF A?
• PROVE THAT THERE ARE 2n
ELEMENTS IN THE CLASS OF ALL SUBSETS OF SET OF n ELEMENTS?
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• WRITE THE FOLLOWING SETS IN SET BUILDER FORM
a) {5,10,15,20}
b) {14,21,28,35,42,…,98}?
• STATE WHETHER FOLLOWING SETS ARE FINITE OR INFINITE?
a) {x:xZ And x>-10}
b){x:xR AND 0<x<1}
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• IF U={1,2,3,4,5,6,7},FIND THE COMPLIMENTOF FOLLOWING SETS?
a) A={1,2,3}
b) B={6,7}
• IF A={a,b,c,d}, B={b,d,e,f} THEN FIND
AB,AB,A-B?
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• IF A={2,4,6,8,10} ,B={1,2,3,4,5,6,7} THEN
FIND (A-B)(B-A)?
• IN A GROUP OF 70 PEOPLE,37 LIKE
COFFEE,52 LIKE TEA AND EACH
PERSON LIKES ATLEAST ONE OF THE
TWO DRINKS.HOW MANY LIKE BOTH
COFFEE AND TEA?
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TESTTEST
SET –A
Q1.WRITE DOWN ALL THE SUBSETS OF
{1/2,1,}?
Q2.IF A={1,2,3,(a,b),c} FIND THE POWER SET OF A?
Q3.IF A’B=U,SHOW THAT AB?
Q4.IN A GROUP OF 75 PEOPLE 30 LIKE FOOTBALL,15 LIKE BOTH HOCKEY AND FOOTBALL.HOW MANY LIKE HOCKEY?
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SET-B
Q1.THERE ARE 210 MEMBERS IN A CLUB 100 OF THEM DRINK
TEA AND 65 DRINK TEA BUT NOT COFFEE.FIND
a) HOW MANY DRINK COFFEE.
b) HOW MANY DRINK COFFEE BUT NOT TEA?
Q2.IF B’A’,SHOW THAT A B?
Q3.PROVE THAT A(BC)=(AB)C