GATE CS 2020 Set 1 — Question 49
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Engineering Mathematics → Mathematical Logic → Predicate Logic & Quantifiers
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Question
Which one of the following predicate formulae is NOT logically valid?Note that is a predicate formula without any free occurrence of .
Correct answer
(C) ∀ x(p(x) → W) ≡ ∀ x p(x) → W
Solution
We analyze the validity of each equivalence given that does not contain any free occurrence of .Option (A):
Since is independent of , the universal quantifier distributes over disjunction. This is a valid equivalence (Prenex Normal Form rule).Option (B):
Since is independent of , the existential quantifier distributes over conjunction. This is a valid equivalence.Option (C):
Let's expand the implication .
LHS:
RHS:
The LHS requires to be true for all (or true), while the RHS only requires to be true for some (or true). These are not equivalent. The correct identity is . Thus, this option is NOT logically valid.Option (D):
LHS:
RHS:
The LHS and RHS are identical. This is a valid equivalence.Therefore, option (C) is the correct answer.
Since is independent of , the universal quantifier distributes over disjunction. This is a valid equivalence (Prenex Normal Form rule).Option (B):
Since is independent of , the existential quantifier distributes over conjunction. This is a valid equivalence.Option (C):
Let's expand the implication .
LHS:
RHS:
The LHS requires to be true for all (or true), while the RHS only requires to be true for some (or true). These are not equivalent. The correct identity is . Thus, this option is NOT logically valid.Option (D):
LHS:
RHS:
The LHS and RHS are identical. This is a valid equivalence.Therefore, option (C) is the correct answer.
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