Definition:Non-symmetric Relation

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

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


An example of a non-symmetric relation:

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

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

Also see

  • Results about symmetry of relations can be found here.