How to most quickly check whether a given sentence is logically equivalent to $p↔q$?

Question

This is from a Discrete Mathematics term test:

Q4. Which of the following statements is/are logically equivalent to $p ↔ q$?

(I) $(¬p \lor q) \land (p \lor ¬q)$

(II) $(¬p \land ¬ q) \lor (p \land q)$

(III) $(¬ p \lor ¬ q) \land (p \lor q)$

(IV) $(¬ p \land q) \lor (p \land¬ q)$

The answer is (I) and (II).

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.

I could easily get the first one since $p \iff q \equiv (p \to q) \land (q \to p)$.

However for (II), I cannot think of simple way other than applying distributive law into 4 terms like into $$((¬ p \land ¬ q) \lor p) \land ((¬ p \land¬ q) \lor q),$$ then another round of distributive law to simplify to $$((¬ p \lor p) \land (p \lor ¬ q)) \land ((¬ p \lor q) \land (¬ q \lor q)),$$ 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?

Answer 1

Using double distribution : \begin{align}(\lnot p \land \lnot q)\lor(p\land q) &\equiv (\lnot p \lor p) \land (\lnot p \lor q) \land (\lnot q \lor p) \land (\lnot q \lor q) \\ &\equiv (\lnot p \lor q) \land (\lnot q \lor p) \\ &\equiv p\iff q \end{align}.

Hope that help!

Answer 2

Q4. Which of the following statements is/are logically equivalent to $p ↔ q$?

(I) $(¬p \lor q) \land (p \lor ¬q)$

(II) $(¬p \land ¬ q) \lor (p \land q)$

(III) $(¬ p \lor ¬ q) \land (p \lor q)$

(IV) $(¬ p \land q) \lor (p \land¬ q)$

The answer is (I) and (II). 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, and feels very tedious. Does anyone have a solution to this problem?

In propositional logic, sentence $S_{p,q}$ being logically equivalent to $(p ↔ q)$ means that $$(S_{p,q}↔(p ↔ q))$$ is a tautology. So:

1. first, plug the same value for $p$ and $q$ into each of the four options and eliminate the one(s) that evaluate as False;
2. next, plug different values for $p$ and $q$ into each of the four options and eliminate the one(s) that evaluate as True;
3. the remaining options are precisely the required sentences.