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  "body": "\n## Statement\n\nLet $A$ be a nonempty set, let $B$ be a set and let $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]].\nThen $f$ is [[injective|0PVG1CRAJN]] if and only if $f$ has a [[left inverse|02486SD328]].\n\n## Remarks\n\nThe hypothesis that $A$ is nonempty is necessary. The empty function from $\\varnothing$ to a nonempty set is injective, but there is no function from a nonempty set to $\\varnothing$.\nTo build a left inverse, fix $a_0 \\in A$. Send each $b = f(a)$ to $a$, and send each $b \\notin f(A)$ to $a_0$.\n\n## Explanations\n\n### L1 human\n\nAn injective function loses no information, so you can go back from each value to its unique input.\nThe values that $f$ never takes can go anywhere, and you need one point $a_0$ to send them to.\n",
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    "name": "Injective iff it has a left inverse",
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        "locator": "Left inverses",
        "url": "https://en.wikipedia.org/wiki/Inverse_function"
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  "msc": [
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  "name": "Injective iff it has a left inverse",
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