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 $\mathcal R \ne \varnothing$.

Let $\mathcal R$ be symmetric.

Then from the definition of Asymmetric, $\mathcal R$ is not asymmetric.

Also, from the definition of Non-Symmetric, it is not non-symmetric.

Let $\mathcal R$ be asymmetric.

Then from the definition of Symmetric, $\mathcal R$ is not symmetric.

Also, from the definition of Non-Symmetric, it is not non-symmetric.

If $\mathcal R$ is not symmetric, and $\mathcal R$ is not asymmetric, then from the definition of Non-Symmetric, it is non-symmetric.