# Definition:Asymmetric Relation

## Definition

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

### Definition 1

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

$\tuple {x, y} \in \mathcal R \implies \tuple {y, x} \notin \mathcal R$

### Definition 2

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

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

## Antisymmetric and Asymmetric Relations

Note the difference between:

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

and:

An antisymmetric relation, in which there may be instances of both $\tuple {x, y} \in \mathcal R$ and $\tuple {y, x} \in \mathcal R$ 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.