GATE CS 2022 Q.4: Analytical and Logical Reasoning

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 in the class.

Statement 1 and Statement 3

Statement 1 and Statement 2

Statement 2 and Statement 4

Statement 3 and Statement 4

Official answer and solution

Official answer: A
From: Official GATE 2022 answer key

Solution

Apply the classical square of opposition for categorical propositions.

Let $S$ denote the set of students and $P$ denote the property of being inquisitive. Given that the class is non-empty ( $|S| \ge 1$ ):

Statement 1 is universal affirmative ( $A$ proposition): 'All $S$ are $P$ '.
Statement 3 is universal negative ( $E$ proposition): 'No $S$ is $P$ '.

Statements 1 and 3 are contrary statements. In categorical logic, two contrary statements cannot both be true simultaneously in any non-empty domain. If all students are inquisitive, it cannot simultaneously be true that no student is inquisitive.

Final answer

Option (A)

Common trap

Confusing contrary statements (which cannot both be true) with subcontrary statements (which cannot both be false).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2022 Computer Science and Information Technology (CS)
Question: 4 of 65
Section: General Aptitude

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