1. **State the problem:** We need to analyze the logical formula $$ (p \to (p \wedge q)) \to (p \to q) $$ and determine its truth or validity. 2. **Recall the definitions and rules:** - The implication $p \to q$ is logically equivalent to $\neg p \lor q$. - The conjunction $p \wedge q$ is true only if both $p$ and $q$ are true. - To check if the formula is a tautology, we can use a truth table or logical equivalences. 3. **Rewrite the formula using equivalences:** $$ (p \to (p \wedge q)) \to (p \to q) $$ Using $p \to r \equiv \neg p \lor r$, rewrite inner implications: $$ (\neg p \lor (p \wedge q)) \to (\neg p \lor q) $$ Again, rewrite the outer implication: $$ \neg (\neg p \lor (p \wedge q)) \lor (\neg p \lor q) $$ 4. **Simplify the negation:** $$ \neg (\neg p \lor (p \wedge q)) = p \wedge \neg (p \wedge q) $$ Using De Morgan's law: $$ p \wedge (\neg p \lor \neg q) $$ Distribute: $$ (p \wedge \neg p) \lor (p \wedge \neg q) $$ Since $p \wedge \neg p$ is false, this simplifies to: $$ p \wedge \neg q $$ 5. **Substitute back:** The whole formula becomes: $$ (p \wedge \neg q) \lor (\neg p \lor q) $$ 6. **Simplify the expression:** Group terms: $$ (p \wedge \neg q) \lor \neg p \lor q $$ Rearranged: $$ \neg p \lor q \lor (p \wedge \neg q) $$ 7. **Analyze the expression:** - If $\neg p$ is true, the formula is true. - If $q$ is true, the formula is true. - If $p$ is true and $q$ is false, then $p \wedge \neg q$ is true, so the formula is true. Thus, in all cases, the formula evaluates to true. **Final conclusion:** The formula $$ (p \to (p \wedge q)) \to (p \to q) $$ is a tautology (always true).