GATE CS 2023 Set 1 — Question 26
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Geetha has a conjecture about integers, which is of the formwhere is a statement about integers, and is a statement about pairs of integers. Which of the following (one or more) option(s) would imply Geetha’s conjecture?
Correct answer
(B) ∀ x ∀ y Q(x, y); (C) ∃ y ∀ x (P(x) ⇒ Q(x, y))
Solution
Geetha's conjecture is .
- Option (B): . Since is a tautology, if is true for all , then is also true for all . Thus, (B) implies .
- Option (C): . Let this be . Then is true. For any , if is true, then is true, which means is true. Thus, is true. So (C) implies .
- Options (A) and (D) are existential statements about and cannot imply a universal statement about all .
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