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  "body": "\n## Definition\n\nLet $A$ be a set.\nAn equivalence relation on $A$ is a [[binary relation|08EJC6Q25H]] $\\sim$ on $A$ that is [[reflexive|0XEGNW851S]], [[symmetric|0N9SA25Z3V]] and [[transitive|0CYRECG2YQ]]. For all $a, b, c \\in A$:\n\n1. $a \\sim a$;\n2. $a \\sim b$ implies $b \\sim a$;\n3. $a \\sim b$ and $b \\sim c$ imply $a \\sim c$.\n\n## Remarks\n\nFor a fixed integer $n \\ge 1$, congruence modulo $n$ on $\\mathbb{Z}$ is an equivalence relation: $a \\sim b$ when $n$ divides $a - b$.\nAn equivalence relation cuts the set into [[equivalence classes|082RMRSY26]], and these form a [[partition|09CCE86A6D]].\nEach [[function|0XHDTE5YMZ]] gives an equivalence relation. See [[equivalence relation induced by a function|0QE5PFJY7V]].\n\n## Explanations\n\n### L1 human\n\nAn equivalence relation says when two objects are \"the same for our purpose\".\nTwo integers are congruent modulo $2$ when they have the same parity.\nThen all even numbers count as one kind, and all odd numbers as another.\n",
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