Symmetric and Transitive Relation is not necessarily Reflexive/Proof 2

Proof
Proof by Counterexample:

Let $S = \Z$ be the set of integers.

Let $\alpha$ be the relation on $S$ defined as:
 * $\forall x, y \in S: x \mathrel \alpha y \iff x = y = 0$

Thus $\alpha$ is symmetric.

Now let $x \mathrel \alpha y$ and $y \mathrel \alpha z$

Then:
 * $x = y = 0, y = z = 0$

and so


 * $x \mathrel \alpha z$

Now let $x = \Z$ such that $x \ne 0$.

Then it is not the case that:
 * $x \mathrel \alpha x$

and so $\alpha$ is not reflexive.

Hence $\alpha$ is both symmetric and transitive but not reflexive.