Existential-universal vs Universal-existential quantifiers

Existential-universal quantifier:

there exists $y$ in $U$ s.t. for every $x$ in $V$, $A(x,y)$

Universal-existential quantifier:

for every $x$ in $V$ there exists $y$ in $U$ s.t. $A(x,t)$

Could you please explain, why existential-universal implies the universal-existential, but not vice versa? Some examples are desirable.

Thank you for your help in advance.

Solution 1:

Suppose $U$ is a set of hats, $V$ is a set of people, and $A(x,y)$ means that person $x$ is willing to wear hat $y$.

Then the first statement says there's a hat that everyone is willing to wear.

The second says that every person is willing to wear at least one of the hats.

Clearly the first statement implies the second.

The second doesn't necessarily imply the first. You might have a bunch of people each of whom owns one hat and wears just that one. There's no hat that everyone will wear.