DEFINITION:SETS 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 SETS TYPES OF REPRESENTATION OF SETS TABULAR OR ROSTER FORM SET-BUILDER OR RULE METHOD
TABULAR 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 METHOD 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 SETS FINITE SET INFINITE SET SINGLETON SET EMPTY SET
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 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 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 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.
SUBSETS THE SET B IS SAID TO BE THE SUBSET LEME OF A IF EVERY ELEMENT OF SET B IS ALSO AN ELEMENT OF A AND WE WRITE IT AS A B OR B A.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 B A.
PROPER 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 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}}
THEOREM: PROVE THAT THERE ARE 2 n ELEMENTS IN THE CLASS OF ALL SUBSETS OF A SET OF n ELEMENTS. PROOF:CONSIDER A SINGLETON SET A={a}.IT HAS TWO POSSIBLE SUBSETS Φ AND {a}. LET CLASS OF ALL SUBSETS OF SET A BE DENOTED AS P(A). THUS, P(A)={Φ,{a}} IF A HAS ONE ELEMENT,THEN P(A) HAS 2 ELEMENTS. CONSIDER SET A={a,b}. IT HAS 4 POSSIBLE SUBSETS Φ,{a},{b},{a,b}
P(A)={Φ,{a},{b},{a,b}} IF A HAS 2 ELEMENTS, THEN P(A) HAS 2 2 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 2 3 ELEMENTS. PROCEEDING THIS WAY WE PROVE THAT, IF A HAS n ELEMENTS THEN P(A) HAS 2 n ELEMENTS.
COMPARABLE SETS TWO SETS A AND B ARE SAID TO BE COMPARABLE IF ONE OF THEM IS SUBSET OF THE OTHER i.e.,EITHER A B OR B A. EXAMPLE: THE SETS {1,3,4,5} AND {1,2,3,4,5,6} ARE COMPARABLE SETS.
DISJOINT 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 SET WHEN 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 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:x U,x A}
EXAMPLES : 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)|x| 20 -20 x 20 ALSO,x IS A MULTIPLE OF -5. B=SET OF ALL MULTIPLES OF -5 WHICH LIES BETWEEN -20 AND 20 ROSTER FORM OF SET B={-20,-15,-10,- 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=5 1,10=5 2,15=5 3,20=5 4 A={x:x=5n,n 4,n N}
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,x N}.
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 DIAGRAM
VENN DIAGRAM OF A U A U
VENN DIAGRAM OF A B A B
UNION 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. A B={x: EITHER x A OR x B}
AB U ABAB
INTERSECTION 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. A B={x:x A AND x B}
ABAB A B
APPLICATION OF SETS IF A AND B ARE NOT DISJOINT SETS THEN n(A B)=n(A)+n(B)-n(A B) IF A AND B ARE DISJOINT SETS THEN n(A B)=n(A)+n(B).
EXAMPLE 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(A B)=65 n(A)=40 n(A B)=10
a)WE KNOW THAT, n(A B)=n(A)+n(B)-n(A B) 65 =40+ n(B)-10 n(B)=35 HENCE,35 PEOPLE LIKE TENNIS. b)NUMBER OF PEOPLE WHO LIKE ONLY TENNIS=n(B)-n(A B)=35-10=25 HENCE,NUMBER OF PEOPLE WHO LIKE TENNIS ONLY AND NOT CRICKET IS 25.
ASSIGNMENT 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 2 n 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:x Z And x>-10} b){x:x R 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 A B,A B,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?
TEST 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 A B? 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 (B C)=(A B) C