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  "body": "\n## Statement\n\nLet $U$ be a set and let $A$ and $B$ be [[subsets|0FE8Q5S42Y]] of $U$. Write $X^c = U \\setminus X$ for the [[complement|0B1FSS1VZW]] of a subset $X$ in $U$.\nThen\n$$(A \\cup B)^{c} = A^{c} \\cap B^{c},$$\n$$(A \\cap B)^{c} = A^{c} \\cup B^{c}.$$\n\n## Remarks\n\nThe laws come from [[De Morgan's laws|09PKJKA6J0]] of logic, applied to $x \\in A$ and $x \\in B$.\nThey hold also for arbitrary families: the complement of $\\bigcup_i A_i$ is $\\bigcap_i A_i^c$, and the complement of $\\bigcap_i A_i$ is $\\bigcup_i A_i^c$.\nWith set differences, $C \\setminus (A \\cup B) = (C \\setminus A) \\cap (C \\setminus B)$ for each set $C$.\n\n## Explanations\n\n### L1 human\n\nAn element is outside $A \\cup B$ exactly when it is outside $A$ and outside $B$.\nAn element is outside $A \\cap B$ exactly when it is outside $A$ or outside $B$.\nA [[Venn diagram|0KCT6DPXJK]] with two circles shows both laws.\n",
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