1. **State the problem:** We need to construct a complete truth table for the compound statement $$[(p \to q) \wedge (r \to \neg q)] \to (p \to \neg r)$$ and then classify it as a tautology, contradiction, or contingency. 2. **Recall the logical connectives and their truth values:** - Implication: $$p \to q$$ is false only when $$p$$ is true and $$q$$ is false; otherwise true. - Negation: $$\neg q$$ is true when $$q$$ is false, and vice versa. - Conjunction: $$A \wedge B$$ is true only when both $$A$$ and $$B$$ are true. - The main statement is an implication, so it is false only when the antecedent is true and the consequent is false. 3. **List all possible truth values for $$p$$, $$q$$, and $$r$$:** There are 3 variables, so $$2^3 = 8$$ rows. | p | q | r | |---|---|---| | T | T | T | | T | T | F | | T | F | T | | T | F | F | | F | T | T | | F | T | F | | F | F | T | | F | F | F | 4. **Calculate intermediate components:** - $$p \to q$$ - $$r \to \neg q$$ - $$p \to \neg r$$ - $$[(p \to q) \wedge (r \to \neg q)]$$ - Entire statement $$[(p \to q) \wedge (r \to \neg q)] \to (p \to \neg r)$$ 5. **Fill the truth table:** | p | q | r | $$p \to q$$ | $$r \to \neg q$$ | $$p \to \neg r$$ | $$[(p \to q) \wedge (r \to \neg q)]$$ | Statement | |---|---|---|------------|----------------|----------------|------------------------------|-----------| | T | T | T | T | F | F | F | T | | T | T | F | T | T | T | T | T | | T | F | T | F | T | F | F | T | | T | F | F | F | T | T | F | T | | F | T | T | T | F | T | F | T | | F | T | F | T | T | T | T | T | | F | F | T | T | T | T | T | T | | F | F | F | T | T | T | T | T | 6. **Analyze the statement column:** The statement is false only when the antecedent is true and the consequent is false. From the table, the statement is true in all rows. 7. **Classification:** Since the statement is true for all possible truth values of $$p$$, $$q$$, and $$r$$, it is a **tautology**. 8. **Logical significance:** This tautology shows a logical relationship between the implications involving $$p$$, $$q$$, and $$r$$. It means that whenever $$p \to q$$ and $$r \to \neg q$$ are both true, it logically guarantees that $$p \to \neg r$$ is also true. **Final answer:** The compound statement is a tautology.