1. **State the problem:** We want to show that the logical statements $p \to q$ and $\neg p \lor q$ are logically equivalent. 2. **Recall the definitions:** - The implication $p \to q$ is false only when $p$ is true and $q$ is false; otherwise, it is true. - The disjunction $\neg p \lor q$ is true if either $\neg p$ (not $p$) is true or $q$ is true. 3. **Construct the truth table:** | $p$ | $q$ | $p \to q$ | $\neg p$ | $\neg p \lor q$ | |-----|-----|-----------|----------|-----------------| | T | T | T | F | T | | T | F | F | F | F | | F | T | T | T | T | | F | F | T | T | T | 4. **Compare columns:** The columns for $p \to q$ and $\neg p \lor q$ are identical for all truth values of $p$ and $q$. 5. **Conclusion:** Since $p \to q$ and $\neg p \lor q$ have the same truth values in all cases, they are logically equivalent. **Final answer:** $p \to q \equiv \neg p \lor q$