1. **State the problems:** We want to verify the logical equivalences: - d. $[p \wedge (q \vee r)] \to [(p \wedge q) \vee (p \wedge r)]$ - e. $[p \vee (q \wedge r)] \to [(p \vee q) \wedge (p \vee r)]$ 2. **Explain:** These are distributive implications in propositional logic, related to distribution of $\wedge$ over $\vee$ and vice versa. 3. **For d:** - The left is $p \wedge (q \vee r)$. - According to distribution, $p \wedge (q \vee r) \equiv (p \wedge q) \vee (p \wedge r)$. - Hence, the implication $[p \wedge (q \vee r)] \to [(p \wedge q) \vee (p \wedge r)]$ is a tautology since both sides are logically equivalent. 4. **For e:** - The left side is $p \vee (q \wedge r)$. - By distribution of $\vee$ over $\wedge$, $p \vee (q \wedge r) \equiv (p \vee q) \wedge (p \vee r)$. - Therefore the implication $[p \vee (q \wedge r)] \to [(p \vee q) \wedge (p \vee r)]$ is also a tautology. 5. **Conclusion:** Both implications are true because they express distributive laws in propositional logic. Final answer: Both d and e are logical tautologies representing distribution laws and thus always hold true.