Geetha has a conjecture about integers, which is of the form

$$\forall x\left( {P(x) \Rightarrow \exists yQ(x,y)} \right)$$,

where P is a statement about integers, and Q is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha's conjecture?

Choose the correct choice(s) regarding the following propositional logic assertion S:

S : ((P ∧ Q)→ R)→ ((P ∧ Q)→ (Q → R))

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?

$$p:\,\,\,x \in \left\{ {8,9,10,11,12} \right\}$$
$$q:\,\,\,x$$ 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 ______________.