# Definition:Asymmetric Relation/Definition 1

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## Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

$\RR$ is **asymmetric** if and only if:

- $\tuple {x, y} \in \RR \implies \tuple {y, x} \notin \RR$

## Antisymmetric and Asymmetric Relations

Note the difference between:

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

and:

- An
**antisymmetric relation**, in which there*may*be instances of both $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$ 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 $a \mathrel \RR b$ 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 $y R x$.

## Also see

- Results about
**asymmetric relations**can be found**here**.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers - 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 - 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations - 1979: John E. Hopcroft and Jeffrey D. Ullman:
*Introduction to Automata Theory, Languages, and Computation*... (previous) ... (next): Chapter $1$: Preliminaries: $1.5$ Relations: Properties of Relations $5)$ - 1996: Winfried Just and Martin Weese:
*Discovering Modern Set Theory. I: The Basics*... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $1$: Pairs, Relations, and Functions