Reflexive and Symmetric Relation is not necessarily Transitive/Proof 1

Theorem

Let $S$ be a set.

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

Let $\alpha$ be both reflexive and symmetric.

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

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

Let:

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

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.

$\blacksquare$