Composite of Reflexive Relations is Reflexive

Theorem
Let $A$ be a set.

Let $\RR$ and $\SS$ be reflexive relations on $A$.

Then their composite $\RR \circ \SS$ is also reflexive.

Proof
Recall the definition of composition of relations:

Hence in this particular context:
 * $\RR \circ \SS := \set {\tuple {x, z} \in A \times A: \exists y \in A: \tuple {x, y} \in \SS \land \tuple {y, z} \in \RR}$

Let $x \in A$ be arbitrary.

By definition of reflexive relation:
 * $\tuple {x, x} \in \SS$
 * $\tuple {x, x} \in \RR$

and so setting $y = z = x$:
 * $\tuple {x, x} \in \set {\tuple {x, z} \in A \times A: \exists y \in A: \tuple {x, y} \in \SS \land \tuple {y, z} \in \RR}$

Hence the result.