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  "body": "\n## Definition\n\nLet $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.\nThe relation $R$ is antisymmetric when, for all $a, b \\in A$, $a \\mathrel{R} b$ and $b \\mathrel{R} a$ imply $a = b$.\n\n## Remarks\n\nThe relation $\\le$ on $\\mathbb{R}$ and the relation $\\subseteq$ on sets are antisymmetric.\nDivisibility is antisymmetric on $\\mathbb{N}$ but not on $\\mathbb{Z}$, because $1 \\mid -1$ and $-1 \\mid 1$.\nAntisymmetric is not the opposite of symmetric: equality is both symmetric and antisymmetric.\n\n## Explanations\n\n### L1 human\n\nIf $a \\le b$ and $b \\le a$, then $a = b$.\nSo for $\\le$, a two-way relation between two elements forces them to be the same.\n",
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