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  "body": "\n## Statement\n\nLet $A$ be a set.\nSend each [[equivalence relation|0SGTMBEWRF]] $\\sim$ on $A$ to its set of [[equivalence classes|082RMRSY26]] $A/{\\sim}$.\nThis map is a [[bijection|0SNQEF7Z53]] from the set of equivalence relations on $A$ to the set of [[partitions|09CCE86A6D]] of $A$.\nIts inverse sends a partition $P$ to the relation $\\sim_P$ with $a \\sim_P b$ when $a$ and $b$ are in the same block of $P$.\n\n## Remarks\n\nThe relation $\\sim_P$ is an equivalence relation, and its classes are the blocks of $P$.\nA smaller equivalence relation, one that relates fewer pairs, gives a finer partition.\nThe number of partitions of a set with $n$ elements is the Bell number $B_n$. For example, $B_3 = 5$.\n\n## Explanations\n\n### L1 human\n\nAn equivalence relation and a partition are two descriptions of the same thing.\nFrom the relation you get the boxes, and from the boxes you get the relation \"is in the same box as\".\n",
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