Union of Transitive Relations Not Always Transitive

Theorem
The union of transitive relations is not necessarily itself transitive.

Proof
Proof by counterexample.

Let $$S = \left \{{a, b, c, d}\right\}$$.

Let $$\mathcal{R}_1$$ be the transitive relation $$\left \{{\left({a, b}\right), \left({b, c}\right), \left({a, c}\right)}\right\}$$.

Let $$\mathcal{R}_2$$ be the transitive relation $$\left \{{\left({b, c}\right), \left({b, d}\right), \left({b, d}\right)}\right\}$$.

Then we have that $$\left({a, b}\right) \in \mathcal{R}_1 \cup \mathcal{R}_2$$ and $$\left({b, d}\right) \in \mathcal{R}_1 \cup \mathcal{R}_2$$.

However, $$\left({a, d}\right) \notin \mathcal{R}_1 \cup \mathcal{R}_2$$, and so $$\mathcal{R}_1 \cup \mathcal{R}_2$$ is not transitive.