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.
 A: It looks like you want to define it as follows:

Given sets $X,Y,$ $A\subseteq X\times Y,$ and $B\subseteq Y$ we define $$A\div B:=\bigl\{x\in X\mid\{x\}\times B\subseteq A\bigr\}.$$

As far as I'm aware, there is no standard name for this.
More generally, if you wanted it to work for arbitrary sets (not just subsets of Cartesian products), you could instead proceed as follows:

The domain of a set $A$, denoted $\text{dom}(A),$ is the set of all $x$ such that $\langle x,y\rangle\in A$ for some $y$. Equivalently, $\text{dom}(A)$ is the domain of the largest binary relation contained in $A$, so while the domain of a non-empty set can be empty, the domain of a non-empty binary relation cannot.
Given sets $A,B,$ we then define $$A\div B:=\bigl\{x\in\text{dom}(A)\mid\{x\}\times B\subseteq A\bigr\}.$$

A: 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 $π_1[A]$.
