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 $\{0,1,2,\mathrm{\dots },n-1\}$. 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