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  "body": "\n## Statement\n\nLet $X$ be a finite set. Let $\\mathcal{P} = \\{p_1, \\dots, p_m\\}$ be a set of $m$ properties that each element of $X$ may or may not have.\n\nFor a subset $S \\subseteq [m] = \\{1, \\dots, m\\}$, let $N(S)$ be the number of elements of $X$ that have every property $p_i$ with $i \\in S$. In particular $N(\\varnothing) = |X|$.\n\nThen the number of elements of $X$ that have none of the properties in $\\mathcal{P}$ is\n$$\\sum_{S \\subseteq [m]} (-1)^{|S|} N(S).$$\n\n## Remarks\n\nThis is the sieve form of the [[inclusion–exclusion principle|0FF7726C7P]]. The sizes are [[cardinalities of finite sets|05099E1NMJ]].\n\nTake $A_i$ to be the set of elements of $X$ with property $p_i$. Then $N(S)$ is the size of the intersection of the $A_i$ for $i \\in S$, and the formula counts the complement of the union of the $A_i$ in $X$.\n",
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  "name": "Inclusion–exclusion formula for elements with none of the properties",
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