Symmetric Transitive and Serial Relation is Reflexive

Theorem
Let $\mathcal R$ be a relation which is:
 * symmetric
 * transitive
 * serial.

Then $\mathcal R$ is reflexive.

Thus such a relation is an equivalence.

Proof
Let $S$ be a set on which $\mathcal R$ is a relation which is symmetric, transitive and serial.

As $\mathcal R$ is symmetric:
 * $x \mathcal R y \implies y \mathcal R x$

As $\mathcal R$ is transitive:
 * $x \mathcal R y \land y \mathcal R x \implies x \mathcal R x$

As $\mathcal R$ is serial:
 * $\forall x \in S: \exists y \in S: x \mathcal R y$

Let $x \in S$.

Then

Thus:
 * $\forall x: x \mathcal R x$

and by definition $\mathcal R$ is reflexive.

It follows by definition that such a relation is an equivalence relation.