GATE CS 2025 Set 2 — Question 15
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Let be an arbitrary predicate over the domain of natural numbers.Which ONE of the following statements is TRUE?
Correct answer
(A) (P(0) ∧ (∀ x [P(x) ⇒ P(x+1)])) ⇒ (∀ x P(x))
Solution
The statement in option (A) corresponds to the Principle of Mathematical Induction.For a predicate over the natural numbers (typically starting from 0):
1.Base Case: is true.
2.Inductive Step: For all , if is true, then is true.
If both conditions hold, then is true for all natural numbers . This is exactly what option (A) states: .Analysis of other options:- (B) suggests backward induction from 0, which would imply properties for negative integers, not covering the natural numbers .
- (C) suggests backward induction starting from 1000. This would prove for , but not for .
- (D) suggests forward induction starting from 1000. This would prove for all , but not for .
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