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  "body": "\n## Definition\n\nLet $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.\nThe relation $R$ is transitive when, for all $a, b, c \\in A$, $a \\mathrel{R} b$ and $b \\mathrel{R} c$ imply $a \\mathrel{R} c$.\n\n## Remarks\n\nThe relations $\u003c$, $\\le$ and $=$ on $\\mathbb{R}$ are transitive, and so is divisibility on $\\mathbb{Z}$.\n\"Is a parent of\" is not transitive.\nThe smallest transitive relation that contains $R$ is the [[transitive closure|0PEJH7AC74]] of $R$.\n\n## Explanations\n\n### L1 human\n\nTransitivity lets you chain steps.\nFrom $2 \u003c 5$ and $5 \u003c 9$ you get $2 \u003c 9$.\nIn the graph of a transitive relation, each path of two arrows has a direct arrow as a shortcut.\n",
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