# 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^{} |