Q:

What are the double union ($\Cup$) and double intersection ($\Cap$) Operators?

Finale of THIS.
Unicode says that $\Cup$ and $\Cap$ are double union and intersection, respectively. I was wondering if there was an actual operation that went with these symbols. If not, would these definitions make sense for these operators? As follows:
$$A\Cup B:=\left\{(x,x):x\in(A\cup B)\right\}$$
and
$$A\Cap B:=\left\{(x,x):x\in(A\cap B)\right\}$$
Question
Do these operators exist within Set Theory? Iff not, do they exist anywhere in the realm of mathematics? Is my idea for these two operators logical and useful?

A:

I think a more useful definition would be $A\Cup B:=\{a\cup b\mid a\in A, b\in B\}$, respectively $A\Cap B:=\{a\cap b\mid a\in A, b\in B\}$.
I've never seen the symbol before, but I could think of situations where that might be useful. I'm sure there are situations where the diagonal of $(A\cup B)^2$ (which you're using) is used, but I have no idea, when you'd want to introduce a special symbol for that, and why you'd want to pick one, that looks more like a union than a diagonal, which is usually denoted by $\Delta$...

