Symmetric and Transitive Relation is not necessarily Reflexive/Proof 1

Theorem

Let $S$ be a set.

Let $\alpha \subseteq S \times S$ be a relation on $S$.

Let $\alpha$ be both symmetric and transitive.

Then it is not necessarily the case that $\alpha$ is also reflexive.

Proof

Proof by Counterexample:

Let $S = \set {a, b, c}$.

Let:

$\alpha = \set {\tuple {a, a}, \tuple {a, b}, \tuple {b, a}, \tuple {b, b} }$

By inspection it is seen that $\alpha$ is both symmetric and transitive.

However, we have:

$\neg c \mathrel \alpha c$

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

$\blacksquare$