Relation is Symmetric and Antisymmetric iff Coreflexive

Theorem
Let $S$ be a set.

Let $\RR \subseteq S \times S$ be a relation in $S$.

Then:
 * $\RR$ is both symmetric and antisymmetric


 * $\RR$ is coreflexive.
 * $\RR$ is coreflexive.

Necessary Condition
Let $\RR$ be both symmetric and antisymmetric

Suppose $\RR \not \subseteq \Delta_S$.

Then:
 * $\exists \tuple {x, y} \in \RR: x \ne y$

But then as $\RR$ is symmetric:
 * $\tuple {y, x} \in \RR$

So we have:
 * $\tuple {x, y} \in \RR$

and:
 * $\tuple {y, x} \in \RR$

where $x \ne y$.

Thus $\RR$ is not antisymmetric, contrary to hypothesis.

So our assumption that $\RR \not \subseteq \Delta_S$ is false.

That is:
 * $\RR \subseteq \Delta_S$

The result follows by definition of coreflexive.

Sufficient Condition
Let $\RR$ be coreflexive.

Then:

and so by definition $\RR$ is symmetric.

Let $\tuple {x, y} \in \RR$ and $\tuple {y, x} \in \RR$.

From the above, this can happen only when $x = y$.

That is, by definition, $\RR$ is antisymmetric.

So $\RR$ is both symmetric and antisymmetric.

Hence the result.