Question

'When the teacher is in the room, all students stand silently.'
If the above statement is true, which one of the following statements is not necessarily true?

• A. If any student is not standing silently, then the teacher is not in the room.
• B. When the teacher is in the room, all students are silent.
• C. If all students are standing, then the teacher is in the room.
• D. When the teacher is in the room, all students are standing.

Hint:

A statement "if P then Q" does not guarantee "if Q then P"; watch for the option that reverses the original logic.

Answer 1

The Correct Option is C

Step 1: Translate the sentence into an if-then form.
Let $T$ stand for "teacher is in the room" and $S$ stand for "all students stand silently". The given fact is the conditional $T \Rightarrow S$. A conditional only makes a promise in one direction, from $T$ towards $S$.

Step 2: Recall the safe moves with a conditional.
From $T \Rightarrow S$, we can safely form the contrapositive $\text{not } S \Rightarrow \text{not } T$, which is always equally true. We can also split $S$ into its two parts, standing and being silent, and each part still follows whenever $T$ holds. What we cannot safely do is flip the arrow around to get $S \Rightarrow T$, since that is a different claim called the converse, and a conditional never guarantees its converse.

Step 3: Sort the four options by which move they use.
Option (A) is exactly the contrapositive of $T \Rightarrow S$, so it is safe and necessarily true. Option (B) only pulls out the silent half of $S$, still following from $T \Rightarrow S$, so it is safe. Option (D) only pulls out the standing half of $S$, again safe for the same reason.

Step 4: Examine option (C) closely.
Option (C) states that students standing forces the teacher to be present, which is the pattern $S \Rightarrow T$. This is the converse of the given rule. Nothing in the original sentence stops students from standing for a completely different reason when the teacher is away, so this converse claim can fail even though the original statement is true.

Step 5: Conclude which option is not guaranteed.
Since a conditional statement never guarantees its converse, option (C) is the one that is not necessarily true.
\[ \boxed{\text{Option (C)}} \]