# Definition:Asymmetric Relation/Also defined as

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## Asymmetric Relation 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 $a \mathrel \RR b$ or $b \mathrel \RR a$ 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 $y R x$.

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Relations - 1964: Steven A. Gaal:
*Point Set Topology*... (previous) ... (next): Introduction to Set Theory: $1$. Elementary Operations on Sets