Consider the following expressions: (i) \(false\) (ii) \(Q\) (iii) \(true\)…

Key insight: From P ∧ (P ⇒ Q) we have both P and P ⇒ Q, so by modus ponens we can infer Q.

• false — Not implied. The premises are satisfiable (they can be true), so they do not entail a contradiction.

• Q — Implied, because from P and P ⇒ Q we derive Q by modus ponens.

• true — Implied. A tautology is true in every model, so it holds in all models of the premises.

• P ∨ Q — Implied, since P is true under the premises, making the disjunction true.

• ¬Q ∨ P — Implied, because P is true under the premises, so this disjunction is true.

Conclusion: Four of the given expressions are logically implied by P ∧ (P ⇒ Q).