# transitive relation

A relation^{} $\mathcal{R}$ on a set $A$ is *transitive ^{}* if and only if
$\forall x,y,z\in A$, $(x\mathcal{R}y\wedge y\mathcal{R}z)\to (x\mathcal{R}z)$.

For example, the “is a subset of” relation $\subseteq $ on any set of sets is transitive. The “less than” relation $$ on the set of real numbers is also transitive.

The “is not equal to” relation $\ne $ on the set of integers is not transitive, because $1\ne 2$ and $2\ne 1$ does not imply $1\ne 1$.

Title | transitive relation |
---|---|

Canonical name | TransitiveRelation |

Date of creation | 2013-03-22 12:15:52 |

Last modified on | 2013-03-22 12:15:52 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 14 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E20 |

Related topic | Reflexive^{} |

Related topic | Symmetric^{} |

Related topic | Antisymmetric |

Defines | transitivity |

Defines | transitive |