How many equivalence relations are there in a set of n elements...?

Question

kinggeorge · Answer

Have you learned much about set partitions?

anonymous · Answer

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

anonymous · Answer

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

anonymous · Answer

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

kinggeorge · Answer

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.

kinggeorge · Answer

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.

anonymous · Answer

ok, but how do you count them?

kinggeorge · Answer

Well, that's a different story.