GATE CS 2017 Set 1 Q.1 (CS Q.1): Propositional and First-Order Logic

The statement $(\neg p) \implies (\neg q)$ is logically equivalent to which of the statements below?

I. $p \implies q$
II. $q \implies p$
III. $(\neg q) \lor p$
IV. $(\neg p) \lor q$

I only

I and IV only

II only

II and III only

Official answer and solution

Official answer: D
From: Official GATE 2017 answer key

Solution

Express the implication $(\neg p) \implies (\neg q)$ in disjunctive form and determine its contrapositive.

Recall the conditional equivalence rule: $A \implies B \equiv \neg A \lor B$ .

Applying this to the given statement:

(\neg p) \implies (\neg q) \equiv \neg(\neg p) \lor (\neg q) \equiv p \lor \neg q \equiv (\neg q) \lor p

This directly matches statement III.

Furthermore, the contrapositive of an implication $A \implies B$ is $\neg B \implies \neg A$ , which is logically equivalent to the original statement. Taking the contrapositive of $(\neg p) \implies (\neg q)$ :

\neg(\neg q) \implies \neg(\neg p) \equiv q \implies p

This directly matches statement II.

Statements I ( $p \implies q \equiv \neg p \lor q$ ) and IV ( $(\neg p) \lor q$ ) are equivalent to each other, but not equivalent to $(\neg p) \implies (\neg q)$ . Hence, only II and III are logically equivalent to the given statement.

Final answer

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2017 Computer Science and Information Technology (CS) Session 1
Question: 1 of 65, printed as CS Q.1
Section: Computer Science and Information Technology

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