First order logic negation of a bracket

Question

I have this problem:

$\neg\left((\forall X)(\exists Y)p(X,Y)\right)$

How do I correctly propagate the negation into the bracket?

like this?

$\neg(\forall X)\neg(\exists Y)\neg p(X,Y)$

Or do I only negate $\forall X$?

Answer 1

You do so as follows: $$\begin{align} \lnot((\forall X)(\exists Y)p(X, Y))&\equiv \lnot (\forall X)(\exists Y)p(X, Y) \\ &\equiv (\exists X)\lnot(\exists Y)p(X, Y) \\ &\equiv (\exists X)(\forall Y)\lnot p(X, Y). \end{align}$$

Answer 2

I should think that would have been one of the first things you learned. If it is NOT true that "for all x, p is true" then we can only say "for some x, p is not true". And if it is NOT true that "there are some x such that p is true" we can say "for all x p is not true".