$\displaystyle

\exists x \forall y p(x,y)

$

This one is saying...that there is a natural number x, such that all natural numbers y are greater than or equal to x?

if I fix 1 as x, then this is true. (assuming 1 is the lowest natural number)

and

$\displaystyle

\exists y \forall x p(x,y)

$

That one is saying, that there is a natural number y, such that all natural numbers x, are less than y?

no matter what value you fix for y, this cannot be true..

Soo, first is true and second is false?

Also... in the instance given (Domain of definition N x N) or even in general, is 0 a natural number?

Thanks so much Dan =]