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  "body": "\n## Definition\n\nLet $(P, \\le_P)$ and $(Q, \\le_Q)$ be [[preorders|0RXS8R6K37]] and let $f \\colon P \\to Q$ be a [[function|0XHDTE5YMZ]].\nThe function $f$ is monotone when, for all $a, b \\in P$, $a \\le_P b$ implies $f(a) \\le_Q f(b)$.\n\n## Remarks\n\nOther names are order-preserving and, for real functions, monotonically increasing or non-decreasing.\nThe function $f$ is antitone when $a \\le b$ implies $f(b) \\le f(a)$.\nThe map $S \\mapsto f(S)$ on [[power sets|0BC7KX51Z8]] is monotone for $\\subseteq$.\n\n## Explanations\n\n### L1 human\n\nA monotone function never reverses the order of two inputs.\nOn $\\mathbb{R}$, $x \\mapsto x^3$ is monotone, and $x \\mapsto x^2$ is not, because $-2 \\le 1$ but $4 \u003e 1$.\n",
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