Use Exercise 68 to find the number of different equivalence relations on a set with n elements, where n is a positive integer not exceeding 10.
Let p(n) denote the number of different equivalence relations on a set with n elements (and by Theorem 2 the number of partitions of a set with n elements). Show that p(n) satisfies the recurrence relation p(n) = Σn−1j=0 C(n − 1, j)p(n − j − 1) and the initial condition p(0) = 1.