definition:sets a set is a well-defined collection of objects. examples: 1.the set of students in a...

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.

REPRESENTATION OF SETSREPRESENTATION OF SETS

TYPES OFREPRESENTATION

OF SETS

TABULAROR

ROSTER FORM

SET-BUILDER OR

RULE METHOD

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}.

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)}

TYPES OF SETSTYPES

OF SETS

FINITE SET

FINITE SET

SINGLETONSET

SINGLETONSET

EMPTYSET

EMPTYSET

INFINITESET

INFINITESET

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.

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,...}

3.SINGLETON SET-A SET CONSISTING OF ONLY ONE ELEMENTIS CALLED A SINGLETON SET.

EXAMPLE:-A={2} IS A SINGLETON SET.

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.

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}

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

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.

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.

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}

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}}

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}

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.

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.

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}.

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.

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}

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}.

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}

• 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}

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}.

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}.

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={׀

VENN DIAGRAMVENN DIAGRAM

VENN DIAGRAM OF AU

A

U

VENN DIAGRAM OF AVENN DIAGRAM OF ABB

A

B

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}

A B

U

AB

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}

AB

AB

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).

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?

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

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.

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?

• 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}

• 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?

• 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?

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?

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