GATE EC 2026 Set 1 — Question 5

Let the original statement be P:
"For each student (S1), there exists another student (S2) in the class such that their scores are at most ten marks away."
This can be written in logical form as: $\forall S_1 \exists S_2 \text{ such that } |\text{score}(S_1) - \text{score}(S_2)| \le 10$.We are told that statement P is false. To find the negation of P, we apply De Morgan's laws to the quantifiers:
$\neg (\forall S_1 \exists S_2 \text{ such that } |\text{score}(S_1) - \text{score}(S_2)| \le 10)$
$\equiv \exists S_1 \neg (\exists S_2 \text{ such that } |\text{score}(S_1) - \text{score}(S_2)| \le 10)$
$\equiv \exists S_1 \forall S_2 \neg ( |\text{score}(S_1) - \text{score}(S_2)| \le 10)$
$\equiv \exists S_1 \forall S_2 \text{ such that } |\text{score}(S_1) - \text{score}(S_2)| > 10$In words, the negation means:
"There exists at least one student (S1) in the class for whom, for all other students (S2), their scores are more than 10 marks away."Let's compare this with the given options:
(A) "For each student, the scores of all the other students are more than 10 marks away." This would be $\forall S_1 \forall S_2 \text{ such that } |\text{score}(S_1) - \text{score}(S_2)| > 10$, which is stronger than the negation and not necessarily true.
(B) "There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away." This matches our derived negation exactly.
(C) "There is exactly one student in the class for whom the scores of some students are more than 10 marks away." This is not the correct negation. The negation requires all other students to be more than 10 marks away from that one student.
(D) "For each student, the score of exactly one other student is more than 10 marks away." This is also not the correct negation.Therefore, option (B) is necessarily true if the original statement is false.The final answer is $\boxed{B}$