1. Assume that R and S are symmetric relations on a set A. Prove that Rns is symmetric.

Answer with explanation:

Suppose, A={ (a, b), (b, a), (c, d), (d, c), (p, q), (q, p), (a, a), (b, b) }

A Relation M is Symmetric, if (p, q) ∈M, then (q, p) ∈M.

⇒It is given that, R and S are symmetric Relation on a Set A.

⇒If R is symmetric, then if (a, b) ∈R, means, (b, a) ∈R. So, R={ (a, b), (b, a) }.

⇒If S is Symmetric, then if (c, d) ∈S, means, (d, c) ∈S. So, S={ (c, d), (d, c) }.

⇒R ∩ S = { (a, b), (b, a), (c, d), (d, a) }

⇒If you will look at the elements of Set, R∩S, there is (a, b) ∈ R∩S, so as (b, a) ∈ R∩S. Also, (c, d) ∈ R∩S, so as (d, a) ∈ R∩S.

Which shows Relation in the set, R∩S is symmetric.