Reflexive and Transitive Relation is not necessarily Symmetric/Proof 1

Proof
Proof by Counterexample:

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

Let:
 * $\alpha = \set {\tuple {a, a}, \tuple {b, b}, \tuple {c, c}, \tuple {a, b}, \tuple {b, c}, \tuple {a, c} }$

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

However, we have:
 * $a \mathrel \alpha b$

but it is not the case that $b \mathrel \alpha a$.

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