a) Consider

The union fails transitivity. ; however,

b) Let be equivalence relations and let .

If both S and T are equivalence relations we know is in both so it is in the intersection for all a.

If then is in S and T, by symmetry is in both S and T, so it is in R.

If (a,b) and (b,c) are in R they are in both S and T, so (a,c) must be in S and T by transitivity of each separately, so (a,c) is in R as well.