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:


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

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 $$\left({a, b}\right) \in \mathcal R \and \left({b, a}\right) \in \mathcal R$$.

Also see

 * Reflexivity
 * Transitivity