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  "body": "\n## Definition\n\nLet $A$ and $B$ be subsets of $\\N$.\nA *many-one reduction* from $A$ to $B$ is a total [[computable function|KZSB5E471K]] $f \\colon \\N \\to \\N$ such that for every $x \\in \\N$,\n$$x \\in A \\iff f(x) \\in B.$$\nThe set $A$ is *many-one reducible* to $B$, written $A \\le_m B$, if such an $f$ exists.\nThe sets $A$ and $B$ are *many-one equivalent*, written $A \\equiv_m B$, if $A \\le_m B$ and $B \\le_m A$.\n\n## Remarks\n\nExample: let $E$ be the set of even numbers and let $f(x) = 0$ for even $x$ and $f(x) = 1$ for odd $x$.\nThen $f$ is total and computable, and $x \\in E$ holds exactly when $f(x) \\in \\set{0}$, so $E \\le_m \\set{0}$.\n\nNon-example: the empty set is not many-one reducible to $\\N$.\nFor any $f$, the point $0 \\notin \\emptyset$ would need $f(0) \\notin \\N$, which is impossible.\n\nThe relation $\\le_m$ is reflexive and transitive.\nIf $A \\le_m B$ and $B$ is a [[computable set|PPGD5KW8JC]], then $A$ is computable.\nIf $A \\le_m B$ and $B$ is [[computably enumerable|TN9G24VABN]], then $A$ is computably enumerable.\n",
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