Let $S$ be a finite set.
Define $\mathscr R(S)$ to be the set of relations on $S$. Define a relation $\mathscr R$ on $\mathscr R(S)$ as follows: $$\mathscr R = \{(\mathscr P, \mathscr Q) \mid \mathscr P, \mathscr Q ∈ \mathscr R(S),\; a\mathscr P\,b\implies a\mathscr Q \,b \;\;∀\;\,a,\, b ∈ S\}.$$
How do I prove that $\mathscr P$ is a subset of $\mathscr Q$? Thanks!