1. **State the problem:** We analyze the logical forms of the statements using the variables: - $W$: It is Wednesday - $S$: I will go shopping - $M$: I will go to a movie 2. **Write the statements in logical form:** (a) If it is Wednesday, then I will go shopping and I will not go to a movie: $$W \to (S \wedge \neg M)$$ (b) I will go shopping only if it is Wednesday and I will not go to a movie: "Only if" means implication from $S$ to $W \wedge \neg M$: $$S \to (W \wedge \neg M)$$ (c) {I will go shopping and not go to a movie} is a sufficient condition for it to be Wednesday: Sufficient condition means: $$(S \wedge \neg M) \to W$$ (d) {I will go shopping and not go to a movie} is a necessary condition for it to be Wednesday: Necessary condition means: $$W \to (S \wedge \neg M)$$ 3. **Compare statements (b)-(d) to (a) and its converse:** - Statement (a) is: $$W \to (S \wedge \neg M)$$ - The converse of (a) is: $$(S \wedge \neg M) \to W$$ 4. **Check equivalences:** - (b) is $$S \to (W \wedge \neg M)$$ which is not equivalent to (a) or its converse. - (c) is $$(S \wedge \neg M) \to W$$ which is the converse of (a). - (d) is $$W \to (S \wedge \neg M)$$ which is exactly (a). 5. **Summary:** - (b) is not equivalent to (a) or its converse. - (c) is equivalent to the converse of (a). - (d) is equivalent to (a). Thus, the logical equivalences are clearly identified.