Definition:Symmetry (Relation)

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

Antisymmetric
Note the difference between:
 * 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:
 * 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.

Linguistic Note
The word symmetry comes from Greek συμμετρεῖν (symmetría) meaning measure together.

Also see

 * Reflexivity
 * Transitivity