Let p and q be two propositions. Consider the following two formulae in propositional logic.

S₁ : (¬p ∧ (p ∨ q)) → q

S₂ : q → (¬p ∧ (p ∨ q))

Which one of the following choices is correct?

The statement $(\neg p) \Rightarrow(\neg q)$ is logically equivalent to which of the statements below?

I. $\quad p \Rightarrow q$

II. $q \Rightarrow p$

III. $(\neg q) \vee p$

IV. $(\neg p) \vee q$

Consider the first-order logic sentence $F: \forall x(\exists y R(x, y))$. Assuming non-empty logical domains, which of the sentences below are implied by $F$?

I. $\quad \exists y(\exists x R(x, y))$

II. $\quad \exists y(\forall x R(x, y))$

III. $\forall y(\exists x R(x, y))$

IV. $\neg \exists x(\forall y \neg R(x, y))$

$$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$
$$q:\,\,\,x$$ is a composite number
$$r:\,\,\,x$$ is a perfect square
$$s:\,\,\,x$$ is a prime number

The integer $$x \ge 2$$ which satisfies $$\neg \left( {\left( {p \Rightarrow q} \right) \wedge \left( {\neg r \vee \neg s} \right)} \right)$$ is ______________.