1. **State the problem:** We want to analyze the logical equivalence of the statement $$[r \to (p \lor q)] \leftrightarrow \sim[r \to (r \lor q)]$$. 2. **Recall the implication rule:** An implication $$a \to b$$ is logically equivalent to $$\sim a \lor b$$. 3. **Rewrite each implication:** - $$r \to (p \lor q) \equiv \sim r \lor (p \lor q)$$ - $$r \to (r \lor q) \equiv \sim r \lor (r \lor q)$$ 4. **Simplify the second implication:** Since $$r \lor q$$ always includes $$r$$, $$r \lor q$$ is true whenever $$r$$ is true, so: $$\sim r \lor (r \lor q) = \sim r \lor r \lor q = \text{True} \lor q = \text{True}$$ 5. **Negate the second implication:** $$\sim[r \to (r \lor q)] = \sim \text{True} = \text{False}$$ 6. **Rewrite the original equivalence:** $$[r \to (p \lor q)] \leftrightarrow \sim[r \to (r \lor q)]$$ becomes $$[\sim r \lor (p \lor q)] \leftrightarrow \text{False}$$ 7. **Equivalence to False means negation:** $$[\sim r \lor (p \lor q)] \leftrightarrow \text{False}$$ is true only when $$\sim r \lor (p \lor q)$$ is false. 8. **Find when $$\sim r \lor (p \lor q)$$ is false:** This is false only if both $$\sim r$$ is false and $$(p \lor q)$$ is false. - $$\sim r$$ false means $$r$$ is true. - $$(p \lor q)$$ false means $$p$$ is false and $$q$$ is false. 9. **Final condition for equivalence to hold:** $$r = \text{True}, p = \text{False}, q = \text{False}$$. **Answer:** The equivalence $$[r \to (p \lor q)] \leftrightarrow \sim[r \to (r \lor q)]$$ holds true only when $$r$$ is true and both $$p$$ and $$q$$ are false.