Statements involving quantifiers

Question

I am confused regarding the following;

If we have a statement, for example, $$\exists_{x} \in X, \forall_{y} \in Y, x + y = 0.$$

Now, I'm wondering if you could just choose $x$ as $-y$, or do you have to pick a specific value for $x$?

Answer 1

The way it is stated there, there needs to be (at least) one value x that fits for every y, so you do have to pick a specific one. Obviously you can find sets and define additions in a way that this holds, however it is very likely that those quantifiers should be swapped, so "for every x there exists a y" and not "there exists a x such that for every y"...

Answer 2

The order of quantifiers matters!

Because $\exists x$ comes before $\forall y$, it means that first you have to choose an $x$, and then your adversary will produce an $y$ (possibly based on which $x$ you chose; he doesn't tell you how he makes his choice) and then you need to be sure that your $x$ will work with that $y$.

If it had been $\forall y$ before $\exists x$, then you could demand to see the adversary's $y$ before you decide which $x$ to use.

Answer 3

With the way that the statement is written, it is claiming there is an $x$ for which $x+y=0$ for ANY $y$. The statement saying that every element in $X$ has an additive inverse in $Y$ would be $$\forall y\in Y\,\exists x\in X\,x+y=0.$$