Question

The number of equivalence relations that can be defined on a set with two elements is

Answer 1

The number of equivalence relations on a finite set is equal to the number of partitions of the set. Now the only partitions of S={a,b} are { {a}, {b} } and S={a,b}. Accordingly there are only two equivalence relations on S which are the diagonal relation {(a,a), (b,b)} and S X S = {(a,a), (a,b), (b,a), (b,b)}