I have two related problems. Prove or disprove:

If $\displaystyle R \subseteq A \times A$ and $\displaystyle S \subseteq A \times A$ are transitive relations then $\displaystyle R \cup S$ is a transitive relation.

If $\displaystyle R \subseteq A \times A$ and $\displaystyle S \subseteq A \times A$ are transitive relations then $\displaystyle R \cap S$ is a transitive relation.

The first, I completely understand, because if you union two transitive relations, the combined relation will still be transitive. (Unless I am way off.)

The second has proven more difficult to prove/disprove. Every example that I have come up with has been "vacuously" transitive. I had never heard of that until the other day and I believe that is why it is proving more difficult for me. If anyone could help with this proof, I would much appreciate it. A good nudge in the right direction?

Thanks!