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ideed Group Title

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

  • one year ago
  • one year ago

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  1. KingGeorge Group Title
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    Have you learned much about set partitions?

    • one year ago
  2. satellite73 Group Title
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    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

    • one year ago
  3. satellite73 Group Title
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    a quick google search finds these are bell numbers https://oeis.org/A000110

    • one year ago
  4. satellite73 Group Title
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    or scroll down to "counting possible partitions" here http://en.wikipedia.org/wiki/Equivalence_relation

    • one year ago
  5. KingGeorge Group Title
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    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.

    • one year ago
  6. KingGeorge Group Title
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    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.

    • one year ago
  7. satellite73 Group Title
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    ok, but how do you count them?

    • one year ago
  8. KingGeorge Group Title
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    Well, that's a different story.

    • one year ago
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