Definition:Asymmetric Relation

Definition

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

$\mathcal R$ is asymmetric iff:

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

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 see

• Symmetry of Relations

• Symmetric Relation
• Antisymmetric Relation
• Non-symmetric Relation

• Results about symmetry of relations can be found here.

Sources

• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.2$