$P = \pi_{1}(P) \times \pi_{2}(P)$?

Question

I’m starting my study of functions, I’m following the book “Proofs and Fundamentals”, by Ethan D. Bloch. This is one of the problems of the book and I’m not sure what would be the solution.

Let $X$ and $Y$ be sets. Let $P \subseteq X \times Y$. Let $\pi_{1}:X\times Y \rightarrow X$ and $\pi_{2}:X \times Y \rightarrow Y$ be the projection maps defined by $\pi_{1}((x,y))=x$ and $\pi_{2}((x,y))=y$ for all $(x,y) \in X \times Y$.

Is it true that $P = \pi_{1}(P) \times \pi_{2}(P)$? Give a proof or a counter-example.

Intuitively, I believe this is true (correct me if I’m wrong please). Although I’m having trouble in formulating the proof for this result. Any ideas? Thank you for your time!

Answer 1

No, this needs not be the case. Consider for instance $P=\{(a,b),(a,c),(d,c)\}$, with $a\ne d$ and $b\ne c$. Then $\pi_1(P)\times \pi_2(P)=\{(a,b),(a,c),(d,c),(d,b)\}$.

Answer 2

No, this is not always true. Take for example $X = Y = \mathbb R$, and $P$ a disk of radius $1$. Then $\pi_1(P) \times \pi_2(P)$ is a square.