GATE 2026 DA Q.24: solution and answer

Official answer: A, B, CSource: Official answer key, GATE 2026 DA (IIT Guwahati)

1. Entailment means every model that makes $X$ true also makes $Y$ true. So no model has $X$ true and $Y$ false.

2. (A) $X\Rightarrow Y$ is equivalent to $\neg X\vee Y$. It is false only when $X$ is true and $Y$ is false, and entailment rules that case out. So the implication is true in every model. Correct.

3. (B) $X\wedge \neg Y$ is true exactly when $X$ is true and $Y$ is false. Entailment says this never happens, so $X\wedge \neg Y$ is FALSE in every model. Correct. This is the same condition as (A), since $\neg (X\Rightarrow Y)\equiv X\wedge \neg Y$.

4. (C) 'If $X$ then $Y$' is the plain-English reading of $X\Rightarrow Y$, so it is correct for the same reason as (A).

5. (D) 'If $Y$ then $X$' is the converse, $Y\Rightarrow X$. Entailment is not symmetric. Take $X=P\wedge Q$ and $Y=P$. Then $X$ entails $Y$. But with $P$ true and $Q$ false, $Y$ is true and $X$ is false, so $Y\Rightarrow X$ fails. So (D) does not follow. Not correct.

6. Hence A, B and C are correct.