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The first assignment in an introductory Group Theory course, we're asked to review basic Set Theory and one of the questions came up which asks us to find the total number of possible equivalence relations given some set $A$ with $n$ elements.

While its trivial that the equivalence relations of $A$ naturally provide a method to construct a partition of $A$ and vice versa, it's not clear to me why there cannot be a unequal numbers of equivalence relations and partitions of $A$. Can someone point me in the right direction?

RobPratt
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surrealcanine81
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    Didn’t you just prove it? There is a bijective equivalence between partitions and equivalence relations. – Randall Sep 15 '22 at 22:56
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    Are you worried that two different equivalence relations determine the same partition? Or that two different partitions determine the same equivalence relation? Or something else? – Karl Sep 16 '22 at 00:34

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In set theory, we say that two sets are of equal size if you can construct a bijection between them, i.e. if we can exactly match each element in one set with a unique element from the other and vice versa. The fact that there is a clear mapping between equivalence relations and partitions is enough to prove their equinumeracy.

ConMan
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