# What are the many ways(and the quickest) to prove that 2 statements are logically equivalent

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?

• $p \iff q$ is true only when both $p$ and $q$ have the same truth values. It is fairly easy to see that option II satisfies that. – Anurag A Sep 28 '18 at 4:57
• So is the general idea to get the possible set of truth values that the original statement allows, for this case (p,q) only can be (T,T) and (F,F) , then do a substitution test to see whether it evaluates as True? – Prashin Jeevaganth Sep 28 '18 at 5:02

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}.
• Only way I know how without using a logic table. Sometimes, this is consider as an equivalence for $\iff$. – Zamarion Sep 28 '18 at 5:12