finite Finite 2001-10-25 11:58:50 2007-12-23 09:59:32 Definition finite set \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{graphicx} \usepackage{xypic} A set $S$ is \emph{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 \emph{cardinality} of $S$. Equivalently, a set $S$ is finite if and only if there is no bijection between $S$ and any proper subset of $S$.