I must be misunderstanding something...
Here's the issue. I'm going to use A and E as my universal and existential quantifiers because I don't know how to type upside down A's and backwards E's.
Q(x,y) is "x + y = x - y"; domain is all integers
For every integer x there is a integer y such that Q(x,y).
My understanding is...this is true when, for every x there is a y for which Q(x,y) is true. Looking at x + y = x - y...if y=0, then x=x. So this whole thing should be true. Correct?
BUT...my other understand is that, when there is an x such that Q(x,y) is false for every y, then the whole thing is false. Is this correct? So if you let x=0....then y=-y, which is always false. So then the whole thing is false.
It can't be both! What am I not getting right?