Some students were not involved in the strike. If the above statement is true, which of the following conclusions is/are logically necessary? 1. Some who were involved in the strike were students. 2. No student was involved in the strike. 3. At least one student was involved in the strike. 4. Some who were not involved in the strike were students. A. 1 and 2 B. 2 C. 4 D. 2 and 3

GATE 2019 · General Aptitude · Statements Follow · medium

Answer: Only conclusion 4 is logically necessary. Answer: C. 4.

1. Formalize the given statement: Given: exists x such that x is a student AND x is not involved in the strike. Let S = set of students, I = set of people involved in the strike. Given: (S minus I) is non-empty, i.e., there exists at least one person in S who is not in I.
2. Evaluate conclusion 4: Conclusion 4: 'Some who were not involved in the strike were students' = exists x such that x is not in I AND x is a student. This is logically identical to the given statement (just the conjunction reordered). So in every model where the given statement is true, conclusion 4 is also true. Conclusion 4 is NECESSARY.
3. Eliminate conclusions 1, 2, 3: Conclusion 2 ('No student was involved') and Conclusion 3 ('At least one student was involved') are contradictory — they cannot both be necessary. The given statement allows models where zero students joined the strike (making 1 and 3 false, 2 true) AND models where some students joined (making 1 and 3 true, 2 false). Since each conclusion fails in at least one valid model, none of 1, 2, 3 is logically necessary.