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  "body": "\n## Definition\n\nLet $f \\colon A \\to B$ be a [[function|0XHDTE5YMZ]].\nThe function $f$ is injective when, for all $a, a' \\in A$, $f(a) = f(a')$ implies $a = a'$.\n\n## Remarks\n\nOther names are one-to-one function and injection.\nThe function $x \\mapsto 2x$ on $\\mathbb{R}$ is injective, and $x \\mapsto x^2$ on $\\mathbb{R}$ is not, because $f(1) = f(-1)$.\nAn injective function with a nonempty domain has a [[left inverse|02486SD328]]. See [[injective iff it has a left inverse|0HYFEEZAEM]].\n\n## Explanations\n\n### L1 human\n\nAn injective function never sends two inputs to the same place.\nTo prove injectivity, assume $f(a) = f(a')$ and deduce $a = a'$.\n",
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