GATE DA 2026 Set 1 — Question 24
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Artificial Intelligence → Propositional Logic → Logical Connectives
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Question
Sentence is said to entail Sentence if whenever is TRUE, also must hold TRUE.Which of the following statements is/are correct if entails ?
Correct answer
(A) X ⇒ Y; (B) X ∧ ¬ Y is FALSE; (C) if X then Y
Solution
Entailment means that in every model (or interpretation) where is true, is also true.Let's analyze the options:
(A) : This notation typically represents the logical implication
(B) is FALSE: By definition, if there is no model where is true and is false, then the sentence is unsatisfiable (always false). This is a standard property of entailment. Correct.
(C) "if then ": This is the natural language equivalent of the implication
(D) "if then ": This corresponds to
(A) : This notation typically represents the logical implication
X → Y. If entails , the implication X → Y is valid (a tautology). Thus, this statement is correct in the context of validity.(B) is FALSE: By definition, if there is no model where is true and is false, then the sentence is unsatisfiable (always false). This is a standard property of entailment. Correct.
(C) "if then ": This is the natural language equivalent of the implication
X → Y. Since the implication holds, this statement is correct.(D) "if then ": This corresponds to
Y → X (the converse). Entailment is not symmetric; does not imply . Incorrect.Therefore, options (A), (B), and (C) are correct.Continue learning with Success Tracker
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