1. The problem is to evaluate the logical expression $$\sim r \to (\sim p \lor \sim q)$$ given the statements: - $p$: The car has been repaired. - $q$: The kids are home. - $r$: We will visit Aunt Tillie. 2. First, understand the meaning of the symbols: - $\sim$ means NOT (negation). - $\lor$ means OR. - $\to$ means implies (if... then...). 3. The expression $$\sim r \to (\sim p \lor \sim q)$$ reads as "If we do NOT visit Aunt Tillie, then either the car has NOT been repaired OR the kids are NOT home." 4. To evaluate the truth of this implication, recall that an implication $A \to B$ is false only when $A$ is true and $B$ is false; otherwise, it is true. 5. Since the problem does not provide truth values for $p$, $q$, or $r$, we cannot assign a definite truth value to the expression without more information. 6. However, we can express the contrapositive of the implication, which is logically equivalent: $$\sim r \to (\sim p \lor \sim q) \equiv \sim (\sim p \lor \sim q) \to r$$ 7. Simplify the contrapositive's antecedent using De Morgan's law: $$\sim (\sim p \lor \sim q) = p \land q$$ 8. So the contrapositive is: $$p \land q \to r$$ 9. This means "If the car has been repaired AND the kids are home, then we will visit Aunt Tillie." 10. Without specific truth values, the expression remains symbolic and cannot be evaluated further. Final answer: The expression $$\sim r \to (\sim p \lor \sim q)$$ is logically equivalent to $$p \land q \to r$$ and its truth depends on the truth values of $p$, $q$, and $r$.