# symmetric relation

A relation^{} $\mathcal{R}$ on a set $A$ is *symmetric ^{}* if and only if
whenever $x\mathcal{R}y$ for some $x,y\in A$ then also $y\mathcal{R}x$.

An example of a symmetric relation on $\{a,b,c\}$ is $\{(a,a),(c,b),(b,c),(a,c),(c,a)\}$. One relation that is not symmetric is $\mathcal{R}=\{(b,b),(a,b),(b,a),(c,b)\}$, because $(c,b)\in \mathcal{R}$ but $(b,c)\notin \mathcal{R}$.

On a finite set^{} with $n$ elements there are ${2}^{{n}^{2}}$ relations,
of which ${2}^{\frac{{n}^{2}+n}{2}}$ are symmetric.

A relation $\mathcal{R}$ that is both symmetric and antisymmetric has the property that $x\mathcal{R}y$ implies $x=y$. On a finite set with $n$ elements there are only ${2}^{n}$ such relations.

Title | symmetric relation |
---|---|

Canonical name | SymmetricRelation |

Date of creation | 2013-03-22 12:15:39 |

Last modified on | 2013-03-22 12:15:39 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 21 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E20 |

Related topic | Reflexive^{} |

Related topic | Transitive3 |

Related topic | Antisymmetric |

Defines | symmetry |

Defines | symmetric |