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?

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.
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