TITLE: Quantifier Notation
QUESTION [14 upvotes]: What's the difference between $\forall \space x \space \exists \space y$ and $\exists \space y \space \forall \space x$ ? I don't believe they mean the same thing even though the quantifiers are attached to the same variable, but I'm having a hard time understanding the difference. Any examples to make the distinction clear would be appreciated.

REPLY [2 votes]: There has been a lot of clear examples given as answers. 
So, not giving any example but trying to make a stronger sense: 
You can regard the quantifiers part of your formula as a bound. Therefore, comparing $\forall x \exists y$ vs $\exists x \forall y$, you should consider the first one as "first bound your formula by $x$ and then relative to $x$, bound it by $y$". For the second one, the matter must be obvious(interchange $x$ and $y$ in the quotation marks). Consequently, these two different ways of bounding are not necessarily the same.