Composite of Transitive Relations is not necessarily Transitive

Theorem
Let $A$ be a set.

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

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

Proof
Recall the definition of composition of relations:


 * Proof by Counterexample


 * Non-Transitive-Composite.png

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

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

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

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

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

We note that:
 * $\tuple {a, b}, \tuple {b, c} \in \RR \circ \SS$

but:
 * $\tuple {a, c} \notin \RR \circ \SS$

Hence, by definition, $\RR \circ \SS$ is not transitive.