Definition:Symmetry (Relation)

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

Symmetric
$$\mathcal{R}$$ is symmetric iff:

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

Note that if $$\mathcal{R}$$ is symmetric, then $$\left({x, y}\right) \in \mathcal{R} \iff \left({y, x}\right) \in \mathcal{R}$$.

This follows because:

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

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

$$\Longrightarrow \left({x, y}\right) \in \mathcal{R} \iff \left({y, x}\right) \in \mathcal{R}$$

Asymmetric
$$\mathcal{R}$$ is asymmetric iff:

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

Antisymmetric
$$\mathcal{R}$$ is antisymmetric iff:

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

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