How to negate quantifiers?

Question

How do I push the negation symbol as far inside the expression? $(\lnot\forall n_1\in \mathbb Z)(\exists n_2\in \mathbb Z)[n_1>n_2]$ do I need to negative $(\exists n_2\in \mathbb Z)$to $(\forall \lnot n_2\in \mathbb Z)$?

Answer 1

You change the $\forall$ to $\exists$ and negate the inside. So For $P$ any statement, which may depend on $x$, we have $\lnot \forall x P \iff \exists x \lnot P$ Intuitively, if we know something is not true for all $x$, there must be some $x$ it is not true of. Similarly $\lnot \exists x P \iff \forall x \lnot P$. Intuitively, if there is no $x$ for which $P$ is true, then for all $x$ $P$ is false.