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

2 Answers 2

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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.

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    it looks like there was some kind of strange edit conflict which caused $p$art of your edit to be undone. Since we agree on the content, I went ahead and fixed it.2011-02-10
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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.