The number of equivalence relations on the set {1, 2, 3, 4} is:

The number of equivalence relations on a set with n elements equals the number of partitions of that set, which is the Bell number B(n).

For the 4-element set {1, 2, 3, 4} we sum the Stirling numbers of the second kind:

B(4) = S(4,1) + S(4,2) + S(4,3) + S(4,4) = 1 + 7 + 6 + 1 = 15.

So there are 15 equivalence relations on {1, 2, 3, 4}.