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.

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