GATE CS 2025 Set 2 Q.15: Propositional and First-Order Logic

Let $P(x)$ be an arbitrary predicate over the domain of natural numbers.

Which ONE of the following statements is TRUE?

$(P(0) \wedge (\forall x [P(x) \Rightarrow P(x + 1)])) \Rightarrow (\forall x \, P(x))$

$(P(0) \wedge (\forall x [P(x) \Rightarrow P(x - 1)])) \Rightarrow (\forall x \, P(x))$

$(P(1000) \wedge (\forall x [P(x) \Rightarrow P(x - 1)])) \Rightarrow (\forall x \, P(x))$

$(P(1000) \wedge (\forall x [P(x) \Rightarrow P(x + 1)])) \Rightarrow (\forall x \, P(x))$

Official answer and solution

Official answer: A
From: Official GATE 2025 answer key

Solution

Identify the logical formulation of the Principle of Mathematical Induction over $\mathbb{N} = \{0, 1, 2, \dots\}$ .

Over the domain of natural numbers $\mathbb{N} = \{0, 1, 2, \dots\}$ , the Principle of Mathematical Induction states that if the base case $P(0)$ holds and the inductive step $\forall x [P(x) \Rightarrow P(x + 1)]$ holds, then $\forall x \, P(x)$ holds. Options (C) and (D) fail to cover $x > 1000$ and $x < 1000$ respectively, and (B) steps downward from $0$ rather than upward.

Final answer

Option (A)

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

Source

Paper: GATE 2025 Computer Science and Information Technology - Session 2 (CS2)
Question: 15 of 65
Section: Computer Science and Information Technology

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