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  "aliases": [
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  "body": "\n## Definition\n\nLet $(A_i)_{i \\in I}$ be a [[family of sets|05Q6G4EH2G]] with a nonempty index set $I$.\nThe intersection of the family is the set\n$$\\bigcap_{i \\in I} A_i = \\{x \\mid \\forall i \\in I, \\, x \\in A_i\\}.$$\nSo $x$ is an element of the intersection when $x \\in A_i$ for each $i \\in I$.\n\n## Remarks\n\nFor $I = \\{1, 2\\}$, this is the [[intersection|0HBRW4E6G1]] $A_1 \\cap A_2$.\nFor an empty index set, the condition is [[vacuously true|0V4QXPGJQK]] for each $x$.\nSo the intersection is defined only relative to a [[universal set|0BPYRMHJ48]] $U$, and it is $U$.\nExample: $\\bigcap_{n \\ge 1} (-1/n, 1/n) = \\{0\\}$ in $\\mathbb{R}$.\n\n## Explanations\n\n### L1 human\n\nThe intersection of a family keeps only the elements that are in every member of the family.\nIt generalizes $A \\cap B$ from two sets to any number of sets.\n",
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