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  "body": "\n## Definition\n\nLet $M = (E, \\mathcal{I})$ be a [[matroid|PX31ZNYS3Z]].\nThe *rank function* of $M$ is the map $r \\colon \\mathcal{P}(E) \\to \\set{0, 1, 2, \\dots}$ given by\n$$r(A) = \\max \\set{\\abs{I} : I \\in \\mathcal{I}, \\ I \\subseteq A}.$$\nThe *rank* of $M$ is $r(E)$.\n\n## Remarks\n\nThe maximum exists because $\\emptyset \\subseteq A$ is independent.\nAll maximal independent subsets of $A$ have $r(A)$ elements, and $r(E)$ is the size of every [[basis|QTSQR7130G]] of $M$.\nA set $A$ is independent if and only if $r(A) = \\abs{A}$.\n\nThe rank function satisfies, for all $A, B \\subseteq E$:\n\n1. $0 \\le r(A) \\le \\abs{A}$.\n2. If $A \\subseteq B$, then $r(A) \\le r(B)$.\n3. $r(A \\cup B) + r(A \\cap B) \\le r(A) + r(B)$ (submodularity).\n\nConversely, every function with these three properties is the rank function of exactly one matroid.\n\nExample: for the matroid of the columns of a matrix, $r(A)$ is the [[rank|0ZC9YTXZRY]] of the submatrix formed by the columns in $A$.\nFor the graphic matroid of a graph with vertex set $V$, $r(A) = \\abs{V} - c(A)$, where $c(A)$ is the number of connected components of the graph $(V, A)$.\n\nNon-example: in the uniform matroid $U_{2,3}$, whose independent sets are the subsets of $\\set{1, 2, 3}$ with at most $2$ elements, the size function $A \\mapsto \\abs{A}$ is not the rank function, because $r(\\set{1, 2, 3}) = 2 \\ne 3$.\n",
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