Reflexive and Symmetric Relation is not necessarily Transitive/Proof 1

Proof
Proof by Counterexample:

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

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

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

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

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

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