# 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:

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

### 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.