Let $P ⊆ X ×Y$. Does $π_1(P)×π_2(P) = P$? Give a proof or a counterexample.

Question

Proofs and fundamentals, exercise 4.2.5. I need your help, maybe it's false. Let $X$ and $Y$ be sets, let $A ⊆ X$ and $B ⊆ Y$ be subsets and let $π_1 : X × Y → X$ and $π_2 : X × Y → Y$ be projection maps. c) Let $P ⊆ X × Y$. Does $π_1(P)×π_2(P) = P$? Give a proof or a counterexample.

Answer 1

Hint:

Note that $\pi_1(P) \times \pi_2(P)$ contains all possible combinations $(x, y)$ with $x \in \pi_1(P)$ and $y \in \pi_2(P)$.

Work an example with $P$ as a couple of elements in your favorite product of sets, computing the projections and their Cartesian product and you'll see what I'm getting at.