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

• Definition:Symmetry (Relation)

• Definition:Symmetric Relation
• Definition:Asymmetric Relation
• Definition:Antisymmetric Relation

• Results about non-symmetric relations can be found here.

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): non-symmetric (of a relation)