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数学中,非对称关系(英语:Asymmetric relation)是二元关系的一种。若集合 上的二元关系 为非对称关系,则对于所有 。换句话说,如果 存在关系,则 不存在关系[1]


一个定义于 上的二元关系 的任何子集。给定 ,我们将 简写为 ,读作“ 存在关系 is related to by )”。

如果对于所有 ,若 ,则 (也就是 ),则我们称 非对称的。以一阶逻辑的形式可以写成:

一个逻辑等价的定义如下:对于所有 中至少有一为假。以一阶逻辑的形式可以写成:

非对称关系的一个例子是定义于实数上的“小于关系”,亦即 。由于当 小于 时, 一定不小于 ,因此 是非对称的。另一方面,“小于等于关系”则不是非对称的,因为当 时, 会同时成立,不符合非对称关系的定义。


非对称关系(Asymmetric)与反对称关系(Antisymmetric)的差异在于:反对称关系容许自反性 可以属于 ,而非对称关系不允许。如上述的“小于等于关系”即是反对称关系的一例。


  • 一个关系为非对称的,当且仅当该关系为反对称非自反[2]
  • 对于一个非对称关系 ,对其施加限制或求其逆关系后,该关系同样是非对称的。例如,由“”定义的关系是非对称关系(若 ),若将集合从实数限缩至整数,该关系同样是非对称的;求该关系的逆关系“”,该逆关系同样是非对称的。
  • 一个递移关系为非对称的,当且仅当该关系为非自反[3]:若存在 使得该关系不是非对称,则由递移性可得到 ,使得该关系同样不是非自反关系。
  • 一个关系为递移性的且非对称的,当且仅当该关系为严格偏序的。
  • 一个非对称关系不一定是全关系。例如,由“严格子集”定义的关系是非对称关系(若 ),但不是全关系()。



  1. ^ Gries, David; Schneider, Fred B., A Logical Approach to Discrete Math, Springer-Verlag: 273, 1993 .
  2. ^ Nievergelt, Yves, Foundations of Logic and Mathematics: Applications to Computer Science and Cryptography, Springer-Verlag: 158, 2002 .
  3. ^ Flaška, V.; Ježek, J.; Kepka, T.; Kortelainen, J. Transitive Closures of Binary Relations I (PDF). Prague: School of Mathematics - Physics Charles University. 2007: 1 [2013-08-20]. (原始内容 (PDF)存档于2013-11-02).  Lemma 1.1 (iv). Note that this source refers to asymmetric relations as "strictly antisymmetric".
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