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Question
If R is a symmetric relation on a set A, then write a relation between R and R−1.
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Solution
Here, R is symmetric on the set A.
Let (a, b) ∈ R
⇒ (b, a) ∈ R [Since R is symmetric]
⇒ (a, b) ∈ R−1 [By definition of inverse relation]
⇒ R ⊂ R−1
Let (x, y) ∈ R−1
⇒ (y, x) ∈ R [By definition of inverse relation]
⇒ (x, y) ∈ R [Since R is symmetric]
⇒ R−1 ⊂ R
Thus, R = R−1
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