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  "body": "\n## Definition\n\nLet $X$ be a set and let $\\mathcal{F}$ be a [[family|0C21Q694AG]] of subsets of $X$.\nA finite set $S \\subseteq X$ is *shattered* by $\\mathcal{F}$ if the set of [[intersections|0HBRW4E6G1]] $\\set{A \\cap S : A \\in \\mathcal{F}}$ is the whole [[power set|00E0YZCERH]] of $S$.\nThe *VC dimension* of $\\mathcal{F}$ is the supremum of $|S|$ over all finite sets $S \\subseteq X$ shattered by $\\mathcal{F}$.\nIf $\\mathcal{F}$ is nonempty, the empty set is shattered, so the VC dimension is a nonnegative integer or $\\infty$.\n\n## Remarks\n\nExample: let $X = \\R$ and let $\\mathcal{F}$ be the family of closed intervals $[a, b]$ with $a \\le b$.\nThe set $\\set{0, 1}$ is shattered: the intersections $\\emptyset$, $\\set{0}$, $\\set{1}$, $\\set{0, 1}$ come from $[5, 6]$, $[0, 0]$, $[1, 1]$ and $[0, 1]$.\nNo set $\\set{p \u003c q \u003c r}$ is shattered, because an interval that contains $p$ and $r$ contains $q$.\nSo the VC dimension is $2$.\n\nNon-example: for the same family, the set $\\set{0, 1, 2}$ is not shattered, since the pattern $\\set{0, 2}$ fails the condition $A \\cap S = \\set{0, 2}$ for every interval $A$.\n\nThe family of all subsets of an infinite set has VC dimension $\\infty$.\nThe half-planes in $\\R^2$ have VC dimension $3$.\n",
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