# Definition:Antisymmetric Relation/Definition 2

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## Definition

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

$\mathcal R$ is **antisymmetric** if and only if:

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

## Also known as

Some sources render this concept as **anti-symmetric relation**.

Some sources (perhaps erroneously) use this definition for an **asymmetric relation**.

## 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 see

- Results about
**symmetry of relations**can be found here.

## Sources

- 1965: E.J. Lemmon:
*Beginning Logic*... (previous) ... (next): $\S 4.5$: Properties of Relations - 1977: Gary Chartrand:
*Introductory Graph Theory*... (previous) ... (next): Appendix $\text{A}.2$: Cartesian Products and Relations - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations