Definition:Symmetry (Relation)

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

Symmetric
$\mathcal R$ is symmetric iff:


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

Asymmetric
$\mathcal R$ is asymmetric iff:


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

Antisymmetric
$\mathcal R$ is antisymmetric iff:


 * $\left({x, y}\right) \in \mathcal R \land \left({y, x}\right) \in \mathcal R \implies x = y$

Note the difference between asymmetric (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 (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).

Non-symmetric
$\mathcal R$ is non-symmetric iff it is neither symmetric nor asymmetric.

An example of a non-symmetric relation:

Let $S = \left\{{a, b, c}\right\}, \mathcal R = \left\{{\left({a, b}\right), \left({b, a}\right), \left({a, c}\right)}\right\}$.


 * $\mathcal R$ is not symmetric, because $\left({a, c}\right) \in \mathcal R$ but $\left({c, a}\right) \notin \mathcal R$.


 * $\mathcal R$ is not asymmetric, because $\left({a, b}\right) \in \mathcal R \land \left({b, a}\right) \in \mathcal R$.

Also see

 * Reflexivity
 * Transitivity