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Set operation:
Union: Let A and B be sets.The union of the sets A and B ,denoted by A ∪ B,is the set that
contains those elements that are either in A or in B ,or in both
A ∪ B={x|xЄA v xЄB }A v xЄA v xЄB }B }
Example: The union of sets {1,3,5}and {1,2,3}is {1,2,3,5}
Intersection: Let A and B be sets.The intersection of the sets A and B, denoted by A ∩ B,is the
set containing those elements in both A and B
A ∩ B={x|xЄA v xЄB }A ∧^ xЄA v xЄB }B }
Example: The intersection of sets {1,3,5}and {1,2,3}is {1,3}
Disjoint sets: Two sets are called disjoint if their intersection is empty
Example: Let A={1,3,5,7,9}andB={2,4,6,8,10}since A ∩ B= ∅^ so A and B are disjoint.
Principle of Inclusion-Exclusion:
ІAA ∪ BІA=ІAAІA+ІABІA-ІAA ∩ B ІA
Difference of two sets: Let A and B be two sets the difference of A and B denoted by A-
B is the set containing those elements that are in A but not in B.The difference of A and B is also
called the complement of B with respect to A.
A−¿B={x|xЄA v xЄB }A ∧ x ∉ B }
Symmetric Difference of two sets
Symmetric difference of two sets A and B, denoted by A^ ⨁^ B^ is the set of those elements in
either A or B but not in both
A ⨁ B ={ x / x ∈ A ⨁ x ∈ B }
A )
=A
Double
complementation Law
AUB=BUA
A ∩ B=B ∩^ A
Commutative Laws
AU(BUC)=(AUB)UC
A ∩ (B ∩ C)=(A ∩ B) ∩ C
Associative Laws
A ∩ (BUC)=(A ∩ B)U(A ∩ C)
AU(B ∩ C)=(AUB) ∩ (AUC)
Distributive Laws
AUB =^
A ∩
B
A ∩ B =^
A U^
B
De Morgan`s Law
Example Prove that
A ∩ B =
A U
B by showing that each set is a subset of other.
Solution: First,suppose that x ∈
A ∩ B.
⟹ x ∉ (^) A ∩ B
⟹ x ∉ (^) A or x ∉ B
⟹ x ∈
A or x ∈^
B
⟹ x (^) ∈
A U^
B
This implies that
A ∩ B⊆
A U
B …….(a)
Now suppose that y (^) ∈
A U^
B.
⟹ y (^) ∈
A or y^ ∈^
B ,
⟹ y ∉ A ∨ y ∉ B
⟹ y ∉ A ∩ B
⟹ (^) y ∈
A ∩ B
This implies that
A U^
B⊆
A ∩ B. ………(b)
Thus by (a) and (b)
A ∩ B =^
A U^
B
Example Use set builder notation and logical equivalences to show that
A ∩ B =
A U
B.
Solution: The following chain of equalities provides a demonstration of this Identity.
A ∩ B ={x │^ x^ ∉^ A^ ∩ B}
={x │⇁ (x ∈ (A ∩ B))}
={x │⇁ (x ∈ A ∧ x ∈ B)}
={x │ x ∉ AVx ∉ B}
={ x │ x (^) ∈
A Vx ∈
B }
A ∩ B ={ x │ x ∈
A U
B }
Exapmle :Use a membership table to show that A ∩ (BUC)=(A ∩ B)U(A ∩ C)
A B C BUC A ∩
(BUC)
(A ∩ B) (A ∩ C) (A ∩ B)U(A ∩ C)
Clearly from the column 5 and column 8
A ∩ (BUC)=(A ∩ B)U(A ∩ C)
Example: show that
AU ( B ∩C )=(^
C U
B ¿ ∩
A
Solution:
AU ( B ∩C )=^
A ∩
¿ ¿) by the first De Morgan’s law
A ∩ (
BU
C ) by the Second De Morgan’s law
BU
C ) ∩
A by the commutative law for intersections
C U
B ) ∩
A by the commutative law for unions
Computer representation of sets:
Q3: Let A={1,2,3,4,5} and B={0,3,6} find
a)AUB b)A ∩ B
c)A-B d)B-A
try yourself
Q4:let A={a,b,c,d,e} and B={a,b,c, d,e,f,g,h} find
a)AUB b)A ∩ B
c)A-B d)B-A
try yourself
Q5:Let A be a set.show that
A =A
Proof:let x ∈
A
⇒ x ∉
A
⇒ (^) x ∈ A
A ⊆ A…(a)
Let y ∈ A
⇒ y ∉
A
⟹ y ∉
A
⟹ A ⊆
A ……(b)
By (a) and (b) we get
A =A
Q8:Find the sets A and B if A-B ={1,5,7,8},B-A={2,10} and A ∩ B={3,6,9}
Solution: We’ll solve this problem by using Venn diagram
Clearly from the figure A={1,3,5,6,7,8,9}and B={2,3,6,9,10}
Q9:Show that if A and B are sets then
AUB =
A ∩
B
a)by showing that each side is a subset of the other side
b)using a membership table
a)let x ∈
AUB
⟹ x ∉ AUB.
⟹ x ∉ A and x ∉ B
⟹ x ∈
A andx ∈
B.
⟹ x ∈
A ∩
B.
AUB ⊆
A ∩
B ……(a)
let y ∈
A ∩
B.
2,
0
⟹ (^) y ∈
A ∩ B ∩C
A U
B U
C ⊆
A ∩B ∩C ….(b)
Combining (a)and (b) we get
A ∩ B∩ C =^
A U
B U
C
b)membership table
A B C A ∩ B ∩C
A ∩ B∩ C
A
B
C
A U
B U
C 1 1 1 1 0 0 0 0
Clearly from column 5
th
and 9
th ´ A ∩ B∩ C =
A U
B U
C
Q13:Show that if A and B are sets then A-B=A ∩
B
Proof:let x ∈ A-B
⟹ x ∈ A but x ∉ B
⟹ x ∈ Aand x ∈
B
⟹ x ∈ A ∩
B
⟹ A-B ⊆ A ∩
B ……(a)
lety ∈ A ∩
B
⟹ y ∈ Aand y ∈
B
⟹ y ∈ A and y ∉ B
⟹ y ∈ A-B
⟹ A ∩
B⊆ A − B ……(b)
Combining (a)and(b)we get A-B=A ∩
B
Q14.Show that if A and B are sets then (A ∩ B)U(A ∩
B )=A
A B A ∩ B
B A ∩
B (A ∩ B)U(A ∩
B )
Clearly from the column 3
rd
and 5
th
(A ∩ B)U(A ∩
B )=A
Q15.Let A,B,C be sets .show that
a)AU(BUC)=(AUB)UC
b)A ∩ (B ∩ C)=(A ∩ B) ∩ C
c) AU(B ∩ C)=(AUB) ∩ ¿AUC)
Solution:try it yourself
Hint:use membership tables
Q16 Let A,B,C be sets .show that(A-B)-C=(A-C)-(B-C)
Solution:
A B C A-B (A-B)-
C
A-C B-C (A-C)-(B-C)
Clearly from column 5
th
and 7
th
(A-B)-C=(A-C)-(B-C)
Q17:let A={0,2,4,6,8,10},B={0,1,2,3,4,5,6}and C={4,5,6,7,8,9,10}.Find
a) A ∩ B∩ C
b)AUBUC
c)( AUB ) ∩ C
d)(^ A^ ∩^ B )UC
solution:try yourself
Q18.Draw the venn diagram for each of the following combinations of the sets A,B,C
