%%% This file is part of PlanetMath snapshot of 2009-01-12
%%% Primary Title: finite
%%% Primary Category Code: 03E10
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%%% Owner: djao
%%% Author(s): djao
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 A set $S$ is \emph{finite} if there exists a \htmladdnormallink{natural number}{http://planetmath.org/encyclopedia/NaturalNumber.html} $n$ and a \htmladdnormallink{bijection}{http://planetmath.org/encyclopedia/BijectiveFunction.html} from $S$ to $n$. Note that we are using the set theoretic definition of natural number, under which the natural number $n$ equals the set $\{0,1,2,\ldots,n-1\}$. If there exists such an $n$, then it is unique, and we call $n$ the \emph{\htmladdnormallink{cardinality}{http://planetmath.org/encyclopedia/Equinumerosity.html}} of $S$.

Equivalently, a set $S$ is finite if and only if there is no bijection between $S$ and any \htmladdnormallink{proper subset}{http://planetmath.org/encyclopedia/ProperSubset.html} of $S$.


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