## ideed 2 years ago How many equivalence relations are there in a set of n elements...?

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1. KingGeorge

Have you learned much about set partitions?

2. satellite73

i think this is not easy it is identical to asking how many partitions there are. maybe i am mistaken, but i don't think this has an obvious answer

3. satellite73

a quick google search finds these are bell numbers https://oeis.org/A000110

4. satellite73

or scroll down to "counting possible partitions" here http://en.wikipedia.org/wiki/Equivalence_relation

5. KingGeorge

It's actually rather straightforward to figure this out. You can make a bijection between the set of set partitions and the set of equivalence relations on a set of n elements by equating partitions to equivalence classes.

6. KingGeorge

To prove this is actually a bijection, just notice that it has an easy inverse (either way you go, just relabel set partitions as equivalence classes and vice versa). Thus, since it has an inverse, it must be bijective.

7. satellite73

ok, but how do you count them?

8. KingGeorge

Well, that's a different story.