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?

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