Definition:Non-symmetric Relation

Definition

Let $\RR \subseteq S \times S$ be a relation in $S$.

$\RR$ is non-symmetric if and only if it is neither symmetric nor asymmetric.

Example

An example of a non-symmetric relation:

Let $S = \set {a, b, c}, \RR = \set {\tuple {a, b}, \tuple {b, a}, \tuple {a, c} }$.

$\RR$ is not symmetric, because $\tuple {a, c} \in \RR$ but $\tuple {c, a} \notin \RR$.
$\RR$ is not asymmetric, because $\tuple {a, b} \in \RR$ and $\tuple {b, a} \in \RR$ also.

Also see

• Results about symmetry of relations can be found here.