GATE Overflow Test Series | Discrete Mathematics | Test 1 | Question: 5

Question

$P$ and $Q$ are two prepositions. Which of the following logical expressions are equivalent?

1. $\neg Q\rightarrow \neg P$
2. $\neg P \vee Q$
3. $P \rightarrow Q$
4. $\neg P \wedge Q$

• A. I, IV
• B. I, II, III
• C. II, IV
• D. I, II, III, IV

Answer 1

I and III are contrapositive statements and so are equivalent.

I and II are also equivalent.

IV is not equal to any of other three.

Correct Answer is B

Answer 2

1. $\neg Q\rightarrow \neg P \equiv \neg(\neg Q) \vee \neg P \equiv Q \vee \neg P \equiv \neg P \vee Q = P \rightarrow Q$
2. $\neg P \vee Q$
3. $P \rightarrow Q$
4. $\neg P \wedge Q$

First three are the same.

So, the correct answer is $(B).$