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.