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  "body": "\n## Definition\n\nLet $R$ be a [[binary relation|08EJC6Q25H]] on a set $A$.\nThe transitive closure of $R$ is the relation $R^{+}$ on $A$ such that $a \\mathrel{R^{+}} b$ when there are $n \\ge 1$ and $x_0, \\dots, x_n \\in A$ with\n$$a = x_0 \\mathrel{R} x_1 \\mathrel{R} x_2 \\cdots x_{n-1} \\mathrel{R} x_n = b.$$\n\n## Remarks\n\nThe relation $R^{+}$ is [[transitive|0CYRECG2YQ]], it contains $R$, and it is a subset of each transitive relation that contains $R$.\nIt is also the intersection of all transitive relations on $A$ that contain $R$.\nExample: the transitive closure of \"is a parent of\" is \"is an ancestor of\".\n\n## Explanations\n\n### L1 human\n\nThe transitive closure adds all the shortcuts.\nIn a graph, $a \\mathrel{R^{+}} b$ means that there is a path of one or more arrows from $a$ to $b$.\n",
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