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, the empty set is both symmetric and asymmetric.

This is why the assumption that $$\mathcal{R} \ne \varnothing$$.


 * Suppose $$\mathcal{R}$$ is symmetric.

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

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


 * Suppose $$\mathcal{R}$$ is 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.