Is $T = \{ ((A,B), A \times B) : A, B \in P(X) \}$ a function?

Question

Given a set $X$, we have a relation $T$ from $P(X)\times P(X)$ to $P(X \times X)$ so defined :

$$T = \{ ((A,B), A \times B) : A, B \in P(X) \}$$

Is $T$ a function? Is it onto or ono to one?

Answer 1

In order to show that $T$ is a function, you need to prove that for each pair of subsets $A$ and $B$ of $X$ there is one and only one subset $U$ of $X\times X$ such that $T(A,B)=U$, that is, $((A,B),U) \in T$.
Given the definition of $T$, it is quite clear that this is so, taking $U=A\times B$, since $((A,B),A\times B)\in T$ and if $((A,B),U)\in T$, then $U=A \times B$.

I'll leave it to you to show that $T$ is one-to-one (straightforward).
About it being onto, I'll leave you a hint: let $$\Delta_X = \{(x,x): x \in X\}.$$ Certainly you have $\Delta_X \in P(X\times X)$. Can you find $A,B\subseteq X$ such that $A\times B=\Delta_X$?