K.RAVINDRA REDDY
Z . P .H .S .
PANDYALAMADUGU
P.ADISESHAIAH
Z . P .H . S .
DANDIKUPPAM
INTRODUCTION
Canter ( 1845-1918).
S
Set Theory was developed by the
famous great mathematician George
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.
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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 = { n3 / 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
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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) = 2n
UNION OF SETS
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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
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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
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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 }
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. . A-B ≠ B-A
So Difference of sets not following
commutative law
Venn-diagram of A-B,
SYMMETRIC DIFFERENCE
OF SETS
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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 = {1,2,3,4------ }, 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 }
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. . 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 = 20 + 44 – n(A∩B)
n(A∩B) = 64-51
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