Composite of Symmetric Relations is not necessarily Symmetric

Theorem

Let $A$ be a set.

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

Then their composite $\RR \circ \SS$ is not necessarily symmetric.

Let:

$\ds A$

$=$

$\ds \set {1, 2, 3}$

$\ds \RR$

$=$

$\ds \set {\tuple {1, 2}, \tuple {2, 1} }$

$\ds \SS$

$=$

$\ds \set {\tuple {2, 3}, \tuple {3, 2} }$

We note that both $\RR$ and $\SS$ are symmetric relations on $A$.

We have by definition of composition of relations that:

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

By inspection, we see that:

$\RR \circ \SS = \set {\tuple {3, 1} }$

demonstrating that $\RR \circ \SS$ is not symmetric.

$\blacksquare$