Definition:Non-symmetric Relation

Definition

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

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

Example

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

• Symmetry of Relations

• Symmetric Relation
• Asymmetric Relation
• Antisymmetric Relation

• Results about symmetry of relations can be found here.