Composite of Connected Relation is not necessarily Connected

Theorem
Let $A$ be a set.

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

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

Proof
Recall the definition of composition of relations:


 * Proof by Counterexample


 * Non-Connected-Composite.png

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

Let $\RR$ be defined as:
 * $\RR = \set {\tuple {a, b}, \tuple {b, c}, \tuple {c, a} }$

Let $\SS$ be defined as $\RR^{-1}$, that is, the inverse of $\RR$:
 * $\SS = \set {\tuple {b, a}, \tuple {c, b}, \tuple {a, c} }$

Both $\RR$ and $\SS$ can be seen to be connected.

Then we have:
 * $\RR \circ \SS = \set {\tuple {a, a}, \tuple {b, b}, \tuple {c, c} }$

and it is immediately apparent that $\RR \circ \SS$ is not connected.