Is placing existential quantifier before universal different from placing universal quantifier before existential?

Question

if I write "there exists x for all y (statement)" vs " for all y there exists an x (statement)" . does it make any difference? does the first statement mean that there is an x which makes the (statement) valid for all the y? does the second statement mean" for any given y, there will be an x which makes the (statement) true? Or do the statements mean same?

Answer 1

Yes, it makes a difference. Take a concrete example:

Suppose your domain is natural numbers.

$\exists x \forall y \ x > y$ means 'There is a number greater than all numbers'. This is false.

But $\forall y \exists x \ x > y$ means 'For every number there is a greater number. This is true.