The GATE Grind

GATE 2026 DA – Question 24

Artificial Intelligence · Logic: propositional and predicate · 1 mark · Multiple select

Sentence $X$ is said to entail Sentence $Y$ if whenever $X$ is $TRUE$, $Y$ also must hold $TRUE$.

Which of the following statements is/are correct if $X$ entails $Y$?

  1. $X \Rightarrow Y$
  2. $X \wedge \neg Y$ is $FALSE$
  3. if $X$ then $Y$
  4. if $Y$ then $X$

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Show answer and explanation

Correct answer: (A) $X \Rightarrow Y$; (B) $X \wedge \neg Y$ is $FALSE$; (C) if $X$ then $Y$

Explanation

If $X$ entails $Y$, there is no case in which $X$ is true and $Y$ is false. So $X \Rightarrow Y$ is always true (A), $X \wedge \neg Y$ is always false (B), and “if $X$ then $Y$” is the same implication in words (C). The reverse, “if $Y$ then $X$”, does not follow, since $Y$ can be true when $X$ is false (D).