Prove or DisproveSolutionconsider that we have a A be arbitr

Prove or Disprove.

consider that we have a A be arbitrary.

as given R and S are equivalence relations on A so they are reflexive,

hence we can say that (a, a) R and (a, a) S.

so (a, a) R S

hence we say that R S is reflexive.

now consider that a, b A are such that (a, b) R S.

so we can say that (a, b) R and (a, b) S.

also if R and S are symmetric, it follows that (b, a) R and (b, a) S.

as (b, a) R S, R S is symmetric.

now suppose a, b, c A are such that (a, b),(b, c) R S.

then we can say that (a, b),(b, c) R and (a, b),(b, c) S.

also if R is transitive, (a, b),(b, c) R implies that (a, c) R.

as S is transitive, (a, b),(b, c) S implies that (a, c) S.

so we can say that (a, c) R and (a, c) S,

so that (a, c) R S.

so we have verified R S is transitive.

as R S is reflexive, symmetric and transitive, and hence R S is an equivalence relation on A