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  "body": "\n## Definition\n\nLet $n \\ge 0$ be an integer, and order $\\set{0, 1}^n$ coordinatewise: $x \\le y$ when $x_i \\le y_i$ for every $i$.\nA [[Boolean function|0X05GS525E]] $f \\colon \\set{0, 1}^n \\to \\set{0, 1}$ is *monotone* if $x \\le y$ implies $f(x) \\le f(y)$.\nThe *Dedekind number* $M(n)$ is the number of monotone Boolean functions $f \\colon \\set{0, 1}^n \\to \\set{0, 1}$.\n\n## Remarks\n\nIdentify $\\set{0, 1}^n$ with the subsets of $\\set{1, \\ldots, n}$.\nA monotone $f$ is determined by the minimal sets $S$ with $f(S) = 1$, and these form an [[antichain|03HF498P1J]].\nSo $M(n)$ is also the number of antichains of subsets of $\\set{1, \\ldots, n}$, including the empty antichain.\n\nExample: $M(2) = 6$.\nThe monotone functions of $x, y$ are $0$, $1$, $x$, $y$, $x \\wedge y$ and $x \\vee y$.\nThe matching antichains are $\\emptyset$, $\\set{\\emptyset}$, $\\set{\\set{1}}$, $\\set{\\set{2}}$, $\\set{\\set{1, 2}}$ and $\\set{\\set{1}, \\set{2}}$.\n\nNon-example: the function $x \\oplus y$, equal to $1$ exactly when $x \\ne y$, is not monotone.\nIndeed $(1, 0) \\le (1, 1)$ but $f(1, 0) = 1 \u003e 0 = f(1, 1)$.\n",
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