Relation Symmetry

Theorem

Every non-null relation has exactly one of these properties: it is either:

symmetric,

asymmetric or

non-symmetric.

Proof

From Relation both Symmetric and Asymmetric is Null, the empty set is both symmetric and asymmetric.

This is why the assumption that $\RR \ne \O$.

Let $\RR$ be symmetric.

Then $\RR$ is not asymmetric by definition.

Also, $\RR$ is not non-symmetric by definition.

Let $\RR$ be asymmetric.

Then $\RR$ is not symmetric by definition.

Also, $\RR$ is not non-symmetric by definition.

If $\RR$ is not symmetric, and $\RR$ is not asymmetric, then $\RR$ is non-symmetric by definition.

$\blacksquare$

Sources

• 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $5$ Properties of Relations