GATE 2024 CS2 Q.12: solution and answer

Official answer: ASource: Official answer key, GATE 2024 CS2 (IISc Bengaluru)

1. Read "X when Y" as "if Y, then X", so the clause after "when" is the hypothesis: $Y\to X$.

2. The hypothesis is "student scores more than 50% marks", which is $q$.

3. The conclusion is "fail grade cannot be given". This is the negation of $p$ ("fail grade can be given"), so it is $\neg p$.

4. Combine them: $q\to \neg p$, which is option (A).

5. Check the other options. (B) $q\to p$ says a student scoring above 50% does get a fail grade, which is the opposite of the statement. (C) $p\to q$ says a fail grade implies a score above 50%, which reverses the direction and drops the negation. (D) $\neg p\to q$ says no fail grade implies a score above 50%, which is also not the statement. A counter-model for (C) and (D): take $p$ true and $q$ false, meaning a fail grade is given to a student at or below 50%. The original statement holds ($q$ is false, so $q\to \neg p$ is true), but $p\to q$ is false. Take $p$ false and $q$ false: the original statement holds, but $\neg p\to q$ is false.

6. Equivalent forms of (A): $q\to \neg p\equiv \neg q\vee \neg p\equiv \neg (p\wedge q)$. A fail grade and a score above 50% never occur together.