1. **State the problem:** We need to verify the logical equivalence of the statement $$(A \to \sim A) \equiv (\sim B \to \sim A)$$. 2. **Recall the implication rule:** An implication $P \to Q$ is logically equivalent to $\sim P \lor Q$. 3. **Rewrite each implication:** - Left side: $A \to \sim A$ is equivalent to $\sim A \lor \sim A$, which simplifies to $\sim A$. - Right side: $\sim B \to \sim A$ is equivalent to $\sim (\sim B) \lor \sim A$, which simplifies to $B \lor \sim A$. 4. **Rewrite the equivalence:** $$(A \to \sim A) \equiv (\sim B \to \sim A)$$ becomes $$\sim A \equiv B \lor \sim A$$ 5. **Analyze the equivalence:** The equivalence $\sim A \equiv B \lor \sim A$ means both sides have the same truth value. 6. **Check truth values:** - If $\sim A$ is true, then $B \lor \sim A$ is true regardless of $B$. - If $\sim A$ is false (i.e., $A$ is true), then $B \lor \sim A$ is true if $B$ is true, false if $B$ is false. Therefore, the equivalence holds only when $B$ is false or $\sim A$ is true. 7. **Conclusion:** The statement $$(A \to \sim A) \equiv (\sim B \to \sim A)$$ is not a tautology; it depends on the truth values of $A$ and $B$. Hence, the two statements are not logically equivalent in all cases.