Question 11

Simplify the premise $A = P \land (P \implies Q)$:
Since $P \implies Q \equiv \neg P \lor Q$, by distribution we have:

Thus, the premise is logically equivalent to $P \land Q$.

An expression $E$ is logically implied by $A$ if the conditional $A \implies E$ is a tautology (that is, whenever $P \land Q$ is true, $E$ must be true):

1. (i) $\text{false}$: $(P \land Q) \implies \text{false}$ is false when $P$ and $Q$ are both true. Not implied.
2. (ii) $Q$: $(P \land Q) \implies Q$ is a tautology. Implied.
3. (iii) $\text{true}$: $(P \land Q) \implies \text{true}$ is always true. Implied.
4. (iv) $P \lor Q$: Since both $P$ and $Q$ are true, $P \lor Q$ is true. Implied.
5. (v) $\neg Q \lor P$: Since $P$ is true, $\neg Q \lor P$ is true. Implied.

Among the 5 expressions, exactly 4 expressions (ii, iii, iv, v) are logically implied by $P \land (P \implies Q)$.