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

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KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.0Have you learned much about set partitions?

satellite73
 3 years ago
Best ResponseYou've already chosen the best response.0i 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

satellite73
 3 years ago
Best ResponseYou've already chosen the best response.0a quick google search finds these are bell numbers https://oeis.org/A000110

satellite73
 3 years ago
Best ResponseYou've already chosen the best response.0or scroll down to "counting possible partitions" here http://en.wikipedia.org/wiki/Equivalence_relation

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.0It'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
 3 years ago
Best ResponseYou've already chosen the best response.0To 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.

satellite73
 3 years ago
Best ResponseYou've already chosen the best response.0ok, but how do you count them?

KingGeorge
 3 years ago
Best ResponseYou've already chosen the best response.0Well, that's a different story.
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