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How would I do this?

Do I separate it into partitions first? or do I order it by $(1,1)(1,2)(1,3)(2,1)$... etc.

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There's a one-to-one correspondence between equivalence relations and partitions, so you can certainly do it by carefully writing down all the ways to partition $\lbrace1,2,3\rbrace$ into subsets.

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By the by, a proof for this 1-1 relation is not too difficult, but you'll find it in Halmos' 'Niave Set Theory' among many other books – tom Oct 6 '11 at 4:37

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