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For a set $X$, let $R(X)$ be the set of equivalence relations on $X$ and let $P(X)$ be the set of partitions of $X$. Prove there is a bijection $\varphi : R(X) \to P(X)$.

Stuck on how to proceed with this. I understand that equivalence relations and partitions are in essence the same. But in terms of writing a proof, I am really lost how to proceed. Any help will be appreciated.

platty
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Nihar Gupta
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    You say that they are in essence the same -- can you show in what way this holds? That is, if I give you an equivalence relation, can you come up with its corresponding partition and vice versa? This would define such a $\varphi$, it then remains to show that it is a bijection. – platty Dec 11 '18 at 22:52
  • In addition to platty's advice, it might help to write down exactly what an element of $R(X)$ and an element of $P(X)$ might look like. – Alex Jones Dec 11 '18 at 23:09
  • See this and this. – Arnaud D. Dec 14 '18 at 16:17

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