1. **State the problem:** Determine if each of the following logical statements is a tautology (always true). 2. **Recall:** A tautology is a formula that is true in every possible interpretation. 3. **Analyze statement (i):** $((\neg P \wedge (P \wedge q)) \to q)$ - Note that $P \wedge q$ is true only if both $P$ and $q$ are true. - $\neg P \wedge (P \wedge q)$ requires $P$ to be both true and false simultaneously, which is impossible. - Therefore, the antecedent $\neg P \wedge (P \wedge q)$ is always false. - An implication with a false antecedent is always true. - Hence, statement (i) is a tautology. 4. **Analyze statement (ii):** $(((p \to q) \wedge (q \to r)) \to (p \to r))$ - This is the transitivity of implication. - If $p$ implies $q$ and $q$ implies $r$, then $p$ implies $r$. - This is a well-known tautology in propositional logic. - Hence, statement (ii) is a tautology. 5. **Analyze statement (iii):** $((\neg q \wedge (p \to q)) \to \neg p)$ - Suppose $\neg q$ is true and $p \to q$ is true. - Since $p \to q$ is true and $q$ is false, $p$ must be false (otherwise $p \to q$ would be false). - Therefore, $\neg p$ is true. - Hence, the implication holds in all cases. - Statement (iii) is a tautology. **Final answers:** - (i) is a tautology. - (ii) is a tautology. - (iii) is a tautology.