Help with statement negation question

Question

Negate each of the following statements:

1. $P(x): x<3$,

2. $Q(x): x^2 > 8$,

3. $R(x): x^2 < 0$

So I've been working on this question for a while and am stuck on the last part, so here's the full question with my answers:

Let $P(x)$, $Q(x)$ and $R(x)$ be the following predicates with domain $\mathbb Z$.

\begin{align}P(x)&: x < 3\\ Q(x)&: x^2 > 8\\ R(x)&: x^2 < 0\end{align}

(a) For each predicate, determine its truth set. My answer is:

\begin{align}P(x)&: \{-\infty,...,-3, -2, -1, 0, 1, 2\}\\ Q(x)&: \{-\infty,...,-4, -3, -2, -1, 3, 4, 5,...,\infty\}\\ R(x)&: \{\; \}\end{align}

(b) Determine whether or not the following statements are true or false.

\begin{align}\forall x\in\mathbb{Z},\;& Q(x) \rightarrow R(x) \tag{1}\\ \forall x\in\mathbb{Z},\;& R(x) \rightarrow P(x) \tag{2}\\ \exists x\in\mathbb{Z}\text{ such that }&P(x) \wedge Q(x) \tag{3}\end{align}

My answer is:

1. False as $R(x)$ is an empty set

2. False as $R(x)$ is an empty set

3. True, as $-3$ is an element of both sets $P(x)$ and $Q(x)$

(c) Negate each of the statements in 4(b). I don't have an answer in this one because I'm not sure how to start it :/ but I think it has something to do with subbing in $(k+1)$ but that might be something else.

Anyway, any help or correction with my current answers would be GREATLY appreciated, thanks!! :)

Answer 1

Hint for (1).

In order to negate e.g. $∀x ∈ \mathbb Z \ (Q(x) → R(x))$ we have to use first the equivalence between $\lnot ∀x$ and $∃x \lnot$ and then the suitable propositional equivalence : $\lnot (p \to q) \equiv (p \land \lnot q)$.

Thus:

$\lnot \ ∀x ∈ \mathbb Z \ (Q(x) → R(x))$

is equivalent to:

$∃x ∈ \mathbb Z \ \lnot (Q(x) → R(x))$

and this in turn to:

$∃x ∈ \mathbb Z \ (Q(x) \land \lnot R(x))$.

---

The same for (2) and (3) is quite similar, exploiting the equivalence: $(p \to \lnot q) \equiv \lnot (p \land q)$.