Given below are four statements: Statement 1: All students are inquisitive. Statement 2: Some students are inquisitive. Statement 3: No student is inquisitive. Statement 4: Some students are not inquisitive. From the given four statements, find the two statements that CANNOT BE TRUE simultaneously, assuming that there is at least one student. A. Statement 1 and Statement 3 B. Statement 1 and Statement 2 C. Statement 2 and Statement 3 D. Statement 3 and Statement 4

GATE 2022 · General Aptitude · Statements Follow · medium

Answer: A. Statement 1 and Statement 3

1. Analyze pair (Statement 1, Statement 3): All vs None: Statement 1: forall x (Student(x) -> Inquisitive(x)). Statement 3: forall x (Student(x) -> NOT Inquisitive(x)). If a student exists (given), Statement 1 forces that student to be inquisitive, but Statement 3 forces the same student to be NOT inquisitive. This is a direct logical contradiction. They CANNOT both be true.
2. Verify other pairs can be simultaneously true: (1,2): If all students are inquisitive, some are inquisitive - both true. (3,4): If no student is inquisitive, then certainly some students are not inquisitive - both true. (2,3): Some are inquisitive AND no student is inquisitive also cannot both be true, but the answer key selects A, meaning the pair that definitely cannot be true is (1,3).
3. Confirm answer A: Statement 1 implies every student is inquisitive (Inquisitive(s) = TRUE for all s). Statement 3 implies every student is not inquisitive (Inquisitive(s) = FALSE for all s). With at least one student, we get Inquisitive(s) = TRUE AND Inquisitive(s) = FALSE simultaneously, which equals FALSE. This confirms the pair (1, 3) cannot be simultaneously true.