This is from a Discrete Mathematics term test question.
Q4. Which of the following statements is/are logically equivalent to $p \iff q$?
Note: $\sim$ means the negation of
(I) $(\sim p \lor q) \land (p \lor \sim q)$
(II) $(\sim p \land \sim q) \lor (p \land q)$
(III) $(\sim p \lor \sim q) \land (p \lor q)$
(IV) $(\sim p \land q) \lor (p \land \sim q)$
The answer is (I) and (II). I could easily get the first one since $p \iff q \equiv (p \to q) \land (q \to p)$.
As this is MCQ, I definitely avoid truth tables. I also want to avoid using a lot of equivalent laws to save time and reduce sources of error. However for (II), I cannot think of simple way other than applying distributive law into 4 terms like into $$((\sim p \land \sim q) \lor p) \land ((\sim p \land \sim q) \lor q)$$ then another round of distributive law to simplify to $$((\sim p \lor p) \land (p \lor \sim q)) \land ((\sim p \lor q) \land (\sim q \lor q))$$, which feels very tedious to me especially since I might mess the ordering up since some of these properties might not be commutative. Does anyone have a solution to this problem?