GATE CS 2015 Set 2 — Question 63
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Which one of the following well formed formulae is a tautology?
Correct answer
(C) [∀ x ∃ y (P(x,y) arrow R(x,y))] rightarrow [∀ x ∃ y (¬ P(x,y) ∨ R(x,y))]
Solution
Option (C) is a tautology because the implication
Option (A) is false; is valid, but the reverse is not.
Option (D) is also valid in standard first-order logic (renaming variables in universal quantification), but (C) is the intended answer based on the definition of implication.
A → B is logically equivalent to . Therefore, is identical to . The formula is of the form , which is always true.Option (A) is false; is valid, but the reverse is not.
Option (D) is also valid in standard first-order logic (renaming variables in universal quantification), but (C) is the intended answer based on the definition of implication.
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