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