Shorthand for quantifying two variables?

Question

Say I have two real variables $x$ and $y$. Can I universally quantify them with $\forall x,y\in\mathbb{R}$ instead of using $\forall x\in\mathbb{R},\forall y\in\mathbb{R}$? Is this correct notation, or is it deemed sloppy by mathematicians?

Answer 1

This is entirely standard shorthand notation. It could be argued that it is slightly unclear whether $x\in\Bbb R$, but mostly this is apparent from context.

If you want to remedy this minor issue without being too verbose, you could, for instance, say $$\forall (x,y)\in\Bbb R^2$$and there is absolutely no more room for any reasonable ambiguity.

Answer 2

There's notation and there's notation!

If you are using notation in the sense of shorthand symbolism helpfully added to mathematicians' English, with "$\forall$" shorthand for "for all", then "$\forall x, y \in \mathbb{R}$" is perfectly acceptable shorthand "for all $x$ and $y$ in the real numbers". There are no strict rules about this sort of thing, no one "correct" way of doing things -- whatever works, works.

If you are (differently) aiming to use notation in the sense of writing formulas of a formalized language, then all bets are off. There will be a determinate answer to what is correct, depending on the precise rules of the language in question. (Something like "$\forall x, y$" might be allowed; usually not, though.)