K.RAVINDRA REDDY Z. P.H.S. PANDYALAMADUGU P.ADISESHAIAH Z. P.H. S. DANDIKUPPAM

INTRODUCTION S Set Theory was developed by the famous great mathematician George Canter ( ).

Definition of SET Set : A set is a group of well defined objects. For representing set we use { } A set can be represented in two ways 1. Roster method : Elements of a set are written in between the parenthesis 2. Set builder form :Elements of a set are described by their common property

ROSTER FORM Write the following sets in ROSTER-FORM. 1.A is the set of odd numbers bellow 20. Ans. A = {1,3,5,7,9,11,13,15,17,19}. 2.B is the set of vowels. Ans. B = {a,e,i,o,u }. 3.P is the set of prime numbers bellow 30. Ans. P = {2,3,5,7,11,13,17,19,23,29}. 4.F is the set of factors of 50. Ans. F = {1,2,5,10,25,50}. 5.M is the set of multiples of 3. Ans. M = {3,6,9,12,15 …………}

SET-BUILDER FORM Write the following sets in SET-BUILDER form. 1.A = {2,4,6,8,10,……..} Ans.A = {x/x is the positive even number}. 2.B = {1,8,27,64,125….} Ans.B = { n 3 / n is positive integer}. 3.F={apple,banana,mango,orange,grapes}. Ans.F = {x / x is a fruit}. 4.W = {sun,mon,tue,wed,thu,fri,Saturday}. Ans. W = {x / x is the week day}. 5.S = {chiru,bala, naga,venki,raviteja}. Ans S = {x / x is a telugu cine film star}.

TYPES OF SETS-1 SINGLETON SET A set which has only one element is called a singleton set. Ex. A = { 2 }. EMPTY SET A set which has no element in it is called an empty set. An emptyset is represented as Ǿ or { }. FINITE SET If the number of elements of a set is countable then the set is called a finite set. Ex.A = {1,2,3,4,5}. INFINITE SET If the number of elements of a set are uncountable then the set is called an infiniteset. Ex.N = {1,2,3,4,5……………………….} SUBSET If every element of the set A is an element of set B then A is called subset of B. It is represented as A ⊂ B. Ex. A = {1,3,5,7,9}, B = {1,2,3,4,5,6,7,8,9,10} here A ⊂ B. NOTE 1.Empty set is subset of every set. NOTE 2.Every set is subset of itself.

TYPES OF SETS-2 SUPER SET If every element of set A is also an element of set B then B is called superset of A. Ex. A = {4,8,12,16,20}, B = {2,4,6,8,10.12,14,16,18,20} here B ⊃ A. PROPER SUBSET If A ⊂ B and A ≠ B then A is called proper subset of B. Ex. A = {1,2,3}, B = {1,2,3,4,5}. IMPROPER SUBSET If A ⊂ B and A = B then A is called an improper subset of B. Ex. A={2,4,6},B={2,4,6}. POWER SET If A is a set, then set of all subsets of A is called powerset of A. Power set of a set A is represented as P(A). NOTE:If n(A) = n then number of elements in P(A) = 2 n

UNION OF SETS A={1,2,3} B={2,3,4} C={4,5,6} Find( AUB),(,BUC),AU(BUC),(AUB)UC AUB={1,2,3}U{2,3,4} AUB={1,2,3,4,} BUA={1,2,3,4} SO AUB=BUA This is called commutative law BUC={2,3,4,}U{4,5,6,} BUC={2,3,4,5,6} AU(BUC)={1,2,3}U{2,3,4,5,6,} AU(BUC)={1,2,3,4,5,6} (AUB)UC={1,2,3,4}U{4,5,6} (AUB)UC={1,2,3,4,5,6} So AU(BUC)=(AUB)UC T This is called Associative law

U

Venn-Diagrams of AUB,BUC,(AUB)UC

INTERSECTION OF SETS A={a,b,c,d,e} B={d,e,f} C={d,e,f,h} A∩B={a,b,c,d,e}∩{d,e,f} A∩B={d,e } B∩A={d,e,f }∩{a,b,c,d,e } B∩A={d,e} B∩C={d,e,f } ∩ {d,e,f,h } B∩C={d,e,f } C∩B={d,e,f,h} ∩ {d,e,f } C∩B={d,e,f } So Intersecton of sets following Commutative law A∩(B∩C) = {a,b,c,d,e } ∩ { d,e,f } A∩(B∩C) = {d,e } (A∩B)∩C = {,d,e } ∩ {d,e,f,h} (A∩B)∩C = {d,e }... A∩(B∩C) = (A∩B)∩C Intersection of SETS following Associate law

Venn diagram of A∩B,

Venn-diagram of A∩(B∩C)

DIFFERENCE OF SETS A={1,3,5,7,9 },B={1,2,3 } A-B={1,3,5,7,9 } – {1,2,3 } A-B= {5,7,9} B-A= {1,2,3 } – {1,3,5,7,9 } B-A = { 2 }... A-B ≠ B-A So Difference of sets not following commutative law

Venn-diagram of A-B,

SYMMETRIC DIFFERENCE OF SETS IF A = {2,4,6,8,10 } B = { 1,2 3, }THEN PROVE THAT A Δ B = (AUB)-(A∩B) = (A-B)U(B-A). Ans: AUB = {2,4,6,8,10 }U {1,2,3 } AUB = {1,2,3,4,6,8,10 } A∩B = {2 } (A U B ) –(A∩B ) = {1,2,3,4,6,8,10 } – { 2 } = {1,3,4,6,8,10 } (A – B ) = {2,4,6,8,10 } – {1,2,3 } ={4,6,8,10 } (B – A) = {1,2,3, } – { 2,4,6,8,10 } = {I,3 } (A – B) U (B – A ) = {4,6,8,10 } U { 1,3 } = {1,3,4,6,8,10 } So A Δ B = ( A –B ) U (B – A ) = (A U B ) – ( A ∩ B ) Here A Δ B is called Symmetri difference of sets

COMPLIMENT OF A SET µ is a universal set A is any set then (µ -A) is called Complement of A this can be represent as A' µ = {1,2,3, }, A = { 2,4 6,--- }, B = {1,3,5 }, find A', B‘, A'∩B', (AUB)'

PROBLEM-1 IF A = { 1,2,3,4,5 } B = {2,3,4,},C={4,5,6,7} THEN PROVE THAT AU(B∩ C) = (AUB) ∩ (AUC). B∩C= {2,3,4} ∩ {4,5,6,7 } B∩C= {4} AU(B∩C)={1,2,3,4,5 } U {4} AU(B∩C)={1,2,3,4,5 } (AUB) = {1,2,3,4,5 } U {2,3,4 } (AUB) ={1,2,3,4,5 } (AUC) = {1,2,3,4,5 } U {4,5,6,7 } (AUC) = {1,2,3,4,5,6,7 } (AUB)∩(AUC) = {1,2,3,4,5,} ∩ {,2,3,4,5,6,7 } (AUB)∩(AUC) = {1,2,3,4,5 }... AU(B∩C) = (AUB) ∩ (AUC)

Venn –diagram of A∩(BUC) A ∩ (BUC) = =

Venn-Diagram of (A∩B)U(A∩C)

U

Venn-Diagram of A',

Venn-Diagram of (A∩B)' (A∩B)'

Venn –Diagram of A' U B'

Venn –Diagram of A- (

PROBLEM-4 PROBLEMS RELATING TO n(A),n(B),n(AUB) & n(A∩B). If n(AUB)= 51, n(A) = 20, n(B) =44 find n(A∩B) Ans: We know n(AUB)=n(A) +n(B)-n(A∩B) 51 = – n(A∩B) n(A∩B) = n(A∩B)= 13

PROBLEM-5 PROBLEMS RELATING TO n(A),n(B),n(C),n(A∩B),n(B∩C),n(C∩A) n(AUBUC) &n(A∩B∩C).

PROBLEM-6 IF A = {…………..} B = {…………..} THEN PROVE THAT A Δ B = (AUB)-(A∩B) = (A-B)U(B-A).

PROBLEM-7 PROVE THAT AU(B∩C) = (AUB) ∩ (AUC).

PROBLEM-8 PROVE THAT A-(BUC) = (A-B) ∩ (A-C). It is enough to prove 1. A-(BUC) ⊂ (A-B)∩ (A-C) 2. (A-B)∩(A-C) ⊂ A-(BUC)

ACKNOWLEDGEMENTS