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  "body": "\n## Statement\n\nLet $\\sim$ be an [[equivalence relation|0SGTMBEWRF]] on a set $A$.\nThen the set $\\{[a] \\mid a \\in A\\}$ of [[equivalence classes|082RMRSY26]] is a [[partition|09CCE86A6D]] of $A$.\n\n## Remarks\n\nEach class $[a]$ is nonempty, because $a \\in [a]$ by reflexivity.\nIf $[a]$ and $[b]$ have a common element, then $a \\sim b$ by symmetry and transitivity, so $[a] = [b]$.\nThe left cosets of a subgroup of a [[group|G5S1M9]] form such a partition. This fact is the key step in [[Lagrange's theorem|3C53TD]].\n\n## Explanations\n\n### L1 human\n\nThe three properties of an equivalence relation are exactly what you need to sort elements into classes.\nReflexivity puts each element in some class.\nSymmetry and transitivity stop two classes from overlapping.\n",
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