# bijection

Let $X$ and $Y$ be sets. A function $f:X\to Y$ that is one-to-one and onto is called a *bijection* or *bijective function* from $X$ to $Y$.

When $X=Y$, $f$ is also called a *permutation ^{}* of $X$.

An important consequence of the bijectivity of a function $f$ is the existence of an inverse function ${f}^{-1}$. Specifically, a function is invertible^{} if and only if it is bijective^{}. Thus if $f:X\to Y$ is a bijection, then for any $A\subset X$ and $B\subset Y$ we have

$f\circ {f}^{-1}(B)$ | $=B$ | ||

${f}^{-1}\circ f(A)$ | $=A$ |

It easy to see the inverse of a bijection is a bijection, and that a composition^{} of bijections is again bijective.

Title | bijection |

Canonical name | Bijection |

Date of creation | 2013-03-22 11:51:35 |

Last modified on | 2013-03-22 11:51:35 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 16 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 03-00 |

Classification | msc 83-00 |

Classification | msc 81-00 |

Classification | msc 82-00 |

Synonym | bijective |

Synonym | bijective function |

Synonym | 1-1 correspondence |

Synonym | 1 to 1 correspondence |

Synonym | one to one correspondence |

Synonym | one-to-one correspondence |

Related topic | Function |

Related topic | Permutation |

Related topic | InjectiveFunction |

Related topic | Surjective |

Related topic | Isomorphism2 |

Related topic | CardinalityOfAFiniteSetIsUnique |

Related topic | CardinalityOfDisjointUnionOfFiniteSets |

Related topic | AConnectedNormalSpaceWithMoreThanOnePointIsUncountable2 |

Related topic | AConnectedNormalSpaceWithMoreThanOnePointIsUncountable |

Related topic | Bo |