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.

Let $S$ be the set of students in the class, where $|S| \ge 1$, and let $I(x)$ denote that student $x$ is inquisitive.

• Statement 1 (Universal Affirmative - A): $\forall x \in S,\, I(x)$ ("All students are inquisitive")

• Statement 2 (Particular Affirmative - I): $\exists x \in S,\, I(x)$ ("Some students are inquisitive")

• Statement 3 (Universal Negative - E): $\forall x \in S,\,

eg I(x)$ ("No student is inquisitive")

• Statement 4 (Particular Negative - O): $\exists x \in S,\,

eg I(x)$ ("Some students are not inquisitive")

In classical logic (the Square of Opposition), universal affirmative (A) and universal negative (E) propositions are contraries. Given that the domain $S$ is non-empty:

• If Statement 1 is true, every student in $S$ is inquisitive.

• If Statement 3 is true, no student in $S$ is inquisitive (i.e., every student is not inquisitive).

Since there is at least one student $s_0 \in S$, Statement 1 requires $I(s_0) = \text{True}$ and Statement 3 requires $I(s_0) = \text{False}$, which is a direct contradiction. Thus, Statement 1 and Statement 3 cannot both be true simultaneously.

Therefore, (A) is the correct option.