Let R and S be relations on a set A. Prove or give a counterexample for each of the following.
(a) If R and S are reflexive, then R ∩ S is reflexive.
(b) If R and S are reflexive, then R ∪ S is reflexive.
(c) If R and S are symmetric, then R ∩ S is symmetric.
(d) If R and S are symmetric, then R ∪ S is symmetric.
(e) If R and S are transitive, then R ∩ S is transitive.
(f ) If R and S are transitive, then R ∪ S is transitive.
(g) If R and S are equivalence relations, then R ∩ S is an equivalence relation.
(h) If R and S are equivalence relations, then R ∪ S is an equivalence relation.