Provide an example of a non-empty transitive relation $r$ on the set $\mathbb N$ such that the relation $r^{\exists}$ defined in the set $\mathcal{P}$($\mathbb N$) by the condition: $$\langle X, Y \rangle \in r^{\exists} \iff \exists x \exists y (x \in X \; \wedge \; y \in Y \; \wedge \langle x, y\rangle \in r)$$ (a) is transitive; (b) is not transitive.
I started with definition of transitive relation. As I understand, $r^{\exists}$ should have 2 conditions: it should be transitive and have condition above (am I right?). But I don't understand how to connect $r$ with $r^{\exists}$?
Any help will be much appreciated.