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Question
If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?
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Solution
R and S are symmetric relations on the set A.
⇒ R ⊂ A×A and S ⊂ A × A
⇒ R ∩ S ⊂ A × A
Thus, R ∩ S is a relation on A.
Let a, b ∈ A such that (a, b) ∈ R ∩ S. Then,
(a, b) ∈ R ∩ S
⇒ (a, b) ∈ R and (a, b) ∈ S
⇒ (b, a) ∈ R and (b, a) ∈ S [Since R and S are symmetric]
⇒ (b, a) ∈ R ∩ S
Thus,
(a, b) ∈ R ∩ S
⇒ (b, a) ∈ R ∩ S for all a, b ∈ A
So, R ∩ S is symmetric on A.
Also,
Let a, b ∈ A such that (a, b) ∈ R ∪ S
⇒ (a, b) ∈ R or (a, b) ∈ S
⇒ (b, a) ∈ R or (b, a) ∈ S [Since R and S are symmetric]
⇒ (b, a) ∈ R ∪ S
So, R ∪ S is symmetric on A.
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