# finite

A set $S$ is finite if there exists a natural number $n$ and a bijection 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 cardinality of $S$.

Equivalently, a set $S$ is finite if and only if there is no bijection between $S$ and any proper subset of $S$.

 Title finite Canonical name Finite Date of creation 2013-03-22 11:53:25 Last modified on 2013-03-22 11:53:25 Owner djao (24) Last modified by djao (24) Numerical id 9 Author djao (24) Entry type Definition Classification msc 03E10 Classification msc 92C05 Classification msc 92B05 Classification msc 18-00 Classification msc 92C40 Classification msc 18-02 Related topic Infinite Defines finite set