TITLE: Confused between Nested Quantifiers
QUESTION [7 upvotes]: I am reading nested quantifiers. I am confused in between two cases,
 1. Existential Quantifier before Universal Quantifier
 2. Universal Quantifier before Existential Quantifier

I would be very thankful if someone highlights the difference between them and also give an example. 
Thanks in advance.

REPLY [5 votes]: Say $M$ is the set of men, $W$ is the set of women, and we will write $w\prec m$ if woman $w$ is the mother of man $m$.
Then this says that every man $m$ has a mother:
$$\forall m\in M: \exists w\in W: w\prec m $$
This says there is a woman $w$ who is every man's mother:
$$\exists w\in W: \forall m\in M: w\prec m $$
The first one is true, but the second is false.