Consider the following relations. R = {(x, y) | x, y are real numbers and x = wy for some rational number w} S=mn,pqm,n,p,q are integer such that n.q ≠ 0  and  qm=pn}Statement-1: S is an equivalence relation but R is not an equivalence relation.Statement-2: R and S both are symmetric.

Since (0, 1) ∈ R but (1, 0) ∈ R, R is not symmetric and hence is not an equivalence relation so statement-2 is false. Next, For the relation S,qm=pn⇒mn=pqThus mn,pq∈S⇒mn=pqwhich shows that S is reflexive and symmetricAgain , mn,pq∈S and pq,rs∈S⇒ mn=pq=rs ⇒mn,rs∈SThus S is transitive and hence S is an equivalence relation. So, statement 1 is true