What is the "reverse" of the cartesian product? Suppose $A = \{a_1,a_2 \}$ and $B = \{b_1,b_2 \}$. Then $A \times B = \{(a_1,b_1), (a_1,b_2), (a_2,b_1), (a_2,b_2) \}$. What is the "reverse" of this operation? In particular, what would $A \div B$ be?
The motivation for this question is from relational algebra. Consider the following two tables:
$$\text{Table A}: \{(s_1,p_1), (s_2,p_1), (s_1,p_2), (s_3,p_1), (s_5,p_3) \}$$ $$\text{Table B}: \{p_1,p_2\}$$
Then $$A \div B = \{s_1 \}$$
In other words, we look at the x-coordinate which has both $p_1$ and $p_2$ as y-coordinates.

Your example actually gives possible definition: $A ÷ B = \{x: \{x\} × B ⊆ A\}$ = the maximal $C$ such that $C × B ⊆ A$. By this you get e.g. $(A × B) ÷ B = A$. Note however that this notation make sense just for binary cartesian product and asks about a subset of $π_1[A]$.
