GATE CS 2016 Set 2 — Question 37
Go beyond PYQs with Success TrackerAI-powered personalised practice and doubt support. Unlimited practice on eligible plans; AI usage limits apply.MCQ+2 / -0.67MediumPredicate Logic & QuantifiersMathematical LogicEngineering Mathematics
Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
Last updated
Question
Which one of the following well-formed formulae in predicate calculus is NOT valid?
Correct answer
(D) ∀ x (p(x) ∨ q(x)) ⇒ (∀ x p(x) ∨ ∀ x q(x))
Solution
Let's analyze each option:(A)
The consequent is equivalent to , which is equivalent to .
Thus, the formula is of the form , which is a tautology (valid).(B)
Existential quantification distributes over disjunction. . This is valid.(C)
If there exists an such that both and are true, then certainly there exists an (the same one) such that is true, and there exists an such that is true. This is valid.(D)
This is NOT valid. Universal quantification does not distribute over disjunction.
Counterexample: Let the domain be integers. Let be " is even" and be " is odd".
is true (every integer is either even or odd).
However, is false (not all are even) and is false (not all are odd).
So the consequent is false while the antecedent is true. The implication fails.
The consequent is equivalent to , which is equivalent to .
Thus, the formula is of the form , which is a tautology (valid).(B)
Existential quantification distributes over disjunction. . This is valid.(C)
If there exists an such that both and are true, then certainly there exists an (the same one) such that is true, and there exists an such that is true. This is valid.(D)
This is NOT valid. Universal quantification does not distribute over disjunction.
Counterexample: Let the domain be integers. Let be " is even" and be " is odd".
is true (every integer is either even or odd).
However, is false (not all are even) and is false (not all are odd).
So the consequent is false while the antecedent is true. The implication fails.
Continue learning with Success Tracker
A step still unclear? Work through it with support
Use Success Tracker to ask about the reasoning, then try another GATE CS question to check your understanding.
AI-powered practice· Unlimited practice on eligible plans
- PYQs with solutions
- Attempt available previous-year questions, then compare your reasoning with the worked solution. Coverage varies by stream.
- Practice that adapts
- Choose a topic, work on weaker areas and bookmark questions to revisit. Your attempts feed your progress tracking.
- AI doubt support
- Ask follow-up questions about a step or concept while practising, instead of stopping at the final answer.
Unlimited practice is available on eligible plans. Free practice and AI usage have limits; check the current plan allowances before choosing.
This page stays readable without an account. AI responses can be wrong; check them against the solution and source material.
More questions on Mathematical Logic
2026 Set 2 Q3A day can only be cloudy or sunny. The probability of a day being cloudy is , independent of…2026 Set 2 Q5‘When it is raining, peacocks dance.’ Based only on this sentence, which one of the following…2026 Set 2 Q8Figures (i) and (ii) represent intercity highway systems. The black dots represent cities and the…2026 Set 1 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…2026 Set 2 Q10An unbiased six-faced dice whose faces are marked with numbers 1, 2, 3, 4, 5, and 6 is rolled twice…