Negation for the formula

Question

Could someone verify my answer please? Thanks

Answer 1

Negating an 'there exists' or 'for all' generally consists of two steps: switch the quantifier, and negate the nested statement.

So, 'there exists $x$ such that $P(x)$' negates to 'for all $x$, not $P(x)$'; similarly, 'for all $x$, $P(x)$' negates to 'there exists $x$ such that not $P(x)$.

So in this case: the negation is "there exists $x\in\mathbb{Z}$ such that not ($p(x)$ and $q(x)$)". We then use DeMorgan's laws to simplify: the statement "not ($p(x)$ and $q(x)$)" simplifies to "(not $p(x)$) or (not $q(x)$)".

So, the last answer is the correct one.