Number of equivalence relations on a set

If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it?

Solution 1:

The maximum number of equivalence classes is $n$ -the identity relation $\{ (x,x) \ | \ x \in X \}$ is an equivalence relation. The number of equivalence relations is the Bell number. The series is in A000110 of OEIS.

Solution 2:

The Stirling numbers of the second kind $S(n,k)$ are by definition the number of ways you can partition some set of $n$ elements into $k$ non-empty and disjoint subsets. But since an equivalence relation may partition your set into any amount of subsets, you need to sum over all possibilities:

$$\sum_{k=1}^n S(n,k)$$

gives you the anwser.

You can read about them here.