# reflexive relation

A relation^{} $\mathcal{R}$ on a set $A$
is *reflexive ^{}* if and only if $a\mathcal{R}a$ for all $a\in A$.

For example, let $A=\{1,2,3\}$. Then $\{(1,1),(2,2),(3,3),(1,3),(3,2)\}$ is a reflexive relation on $A$, because it contains $(a,a)$ for all $a\in A$. However, $\{(1,1),(2,2),(2,3),(3,1)\}$ is not reflexive because it does not contain $(3,3)$.

On a finite set^{} with $n$ elements there are ${2}^{{n}^{2}}$ relations,
of which ${2}^{{n}^{2}-n}$ are reflexive.

Title | reflexive relation |

Canonical name | ReflexiveRelation |

Date of creation | 2013-03-22 12:15:36 |

Last modified on | 2013-03-22 12:15:36 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 17 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E20 |

Related topic | Symmetric^{} |

Related topic | Transitive3 |

Related topic | Antisymmetric |

Related topic | Irreflexive^{} |

Defines | reflexivity |

Defines | reflexive |