# antisymmetric

A relation^{} $\mathcal{R}$ on $A$ is *antisymmetric* iff
$\forall x,y\in A$, $(x\mathcal{R}y\wedge y\mathcal{R}x)\to (x=y)$.
For a finite set^{} $A$ with $n$ elements, the number of possible antisymmetric relations is ${2}^{n}{3}^{\frac{{n}^{2}-n}{2}}$ out of the ${2}^{{n}^{2}}$ total possible
relations.

Antisymmetric is not the same thing as “not symmetric^{}”, as it is possible
to have both at the same time. However, a relation $\mathcal{R}$ that is both
antisymmetric and symmetric has the condition that $x\mathcal{R}y\Rightarrow x=y$.
There are only ${2}^{n}$ such possible relations on $A$.

An example of an antisymmetric relation on $A=\{\circ ,\times ,\star \}$ would be $\mathcal{R}=\{(\star ,\star ),(\times ,\circ ),(\circ ,\star ),(\star ,\times )\}$. One relation that isn’t antisymmetric is $\mathcal{R}=\{(\times ,\circ ),(\star ,\circ ),(\circ ,\star )\}$ because we have both $\star \mathcal{R}\circ $ and $\circ \mathcal{R}\star $, but $\circ \ne \star $

Title | antisymmetric |
---|---|

Canonical name | Antisymmetric |

Date of creation | 2013-03-22 12:15:50 |

Last modified on | 2013-03-22 12:15:50 |

Owner | aoh45 (5079) |

Last modified by | aoh45 (5079) |

Numerical id | 14 |

Author | aoh45 (5079) |

Entry type | Definition |

Classification | msc 03E20 |

Synonym | antisymmetry |

Related topic | Reflexive^{} |

Related topic | Symmetric |

Related topic | ExteriorAlgebra |

Related topic | SkewSymmetricMatrix |