1. **State the problem:** Construct the truth table for the compound statement $ (p \to q) \lor (\neg p \land \neg q) $ and determine if it is a tautology, contradiction, or contingency. 2. **Recall definitions:** - $p \to q$ (implication) is false only when $p$ is true and $q$ is false; otherwise true. - $\neg p$ is the negation of $p$. - $\land$ is logical AND. - $\lor$ is logical OR. 3. **List all possible truth values for $p$ and $q$:** | $p$ | $q$ | |-----|-----| | T | T | | T | F | | F | T | | F | F | 4. **Calculate $p \to q$ for each row:** - Row 1: $T \to T = T$ - Row 2: $T \to F = F$ - Row 3: $F \to T = T$ - Row 4: $F \to F = T$ 5. **Calculate $\neg p$ and $\neg q$ for each row:** - Row 1: $\neg T = F$, $\neg T = F$ - Row 2: $\neg T = F$, $\neg F = T$ - Row 3: $\neg F = T$, $\neg T = F$ - Row 4: $\neg F = T$, $\neg F = T$ 6. **Calculate $\neg p \land \neg q$ for each row:** - Row 1: $F \land F = F$ - Row 2: $F \land T = F$ - Row 3: $T \land F = F$ - Row 4: $T \land T = T$ 7. **Calculate the entire expression $(p \to q) \lor (\neg p \land \neg q)$ for each row:** - Row 1: $T \lor F = T$ - Row 2: $F \lor F = F$ - Row 3: $T \lor F = T$ - Row 4: $T \lor T = T$ 8. **Analyze the results:** The expression is true in rows 1, 3, and 4, but false in row 2. 9. **Conclusion:** Since the expression is not always true or always false, it is a **contingency**. **Final answer:** The statement $ (p \to q) \lor (\neg p \land \neg q) $ is a contingency.