Definition:Asymmetric Relation

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Definition

Let $\mathcal R \subseteq S \times S$ be a relation in $S$.


Definition 1

$\mathcal R$ is asymmetric if and only if:

$\left({x, y}\right) \in \mathcal R \implies \left({y, x}\right) \notin \mathcal R$


Definition 2

$\mathcal R$ is asymmetric if and only if it and its inverse are disjoint:

$\mathcal R \cap \mathcal R^{-1} = \varnothing$


Note the difference between:

An asymmetric relation, in which the fact that $\left({x, y}\right) \in \mathcal R$ means that $\left({y, x}\right)$ is definitely not in $\mathcal R$

and:

An antisymmetric relation, in which there may be instances of both $\left({x, y}\right) \in \mathcal R$ and $\left({y, x}\right) \in \mathcal R$ but if there are, then it means that $x$ and $y$ have to be the same object.


Also defined as

Some sources (possibly erroneously or carelessly) gloss over the differences between this and the definition for an antisymmetric relation, and end up using a definition for antisymmetric which comes too close to one for asymmetric.

An example is 1964: Steven A. Gaal: Point Set Topology:

[After having discussed antireflexivity] ... antisymmetry expresses the additional fact that at most one of the possibilities $a \mathop {\mathcal R} b$ or $b \mathop {\mathcal R} a$ can take place.


Some sources specifically define a relation as anti-symmetric what has been defined on $\mathsf{Pr} \infty \mathsf{fWiki}$ as asymmetric

From 1955: John L. Kelley: General Topology: Chapter $0$: Relations:

... the relation $R$ is anti-symmetric iff it is never the case that both $x R y$ and $y R x$.


Also see

  • Results about symmetry of relations can be found here.