'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?
If the above statement is true, which one of the following statements is not necessarily true?
Show Hint
- If any student is not standing silently, then the teacher is not in the room.
- When the teacher is in the room, all students are silent.
- If all students are standing, then the teacher is in the room.
- When the teacher is in the room, all students are standing.
The Correct Option is C
Solution and Explanation
The statement is "When the teacher is in the room, all students stand silently." If we let \(P\) be "the teacher is in the room" and \(Q\) be "all students stand silently", the statement says
\[ P \Rightarrow Q \]
This means whenever \(P\) is true, \(Q\) must also be true. It says nothing about what happens when \(P\) is false.
Step 2: Check option (A).
Option (A) says: if any student is not standing silently, then the teacher is not in the room. This is the contrapositive of \(P \Rightarrow Q\), that is \(\text{not } Q \Rightarrow \text{not } P\). A contrapositive is always logically equivalent to the original statement, so this must necessarily be true.
Step 3: Check option (B).
Option (B) says: when the teacher is in the room, all students are silent. "Standing silently" already includes being silent as part of its meaning, so if \(Q\) is guaranteed, being silent is guaranteed too. This follows directly from \(P \Rightarrow Q\) and is necessarily true.
Step 4: Check option (D).
Option (D) says: when the teacher is in the room, all students are standing. Standing is also part of what "stand silently" means, so this too follows directly from \(P \Rightarrow Q\) and is necessarily true.
Step 5: Check option (C).
Option (C) says: if all students are standing, then the teacher is in the room. This argues backwards, from students standing to the teacher being present. The original statement only tells us that the teacher being present forces students to stand silently. It never rules out students standing for some unrelated reason, such as a fire drill or a different visitor, even when the teacher is absent. So this reversed claim is not guaranteed.
Final Answer:
Option (C) commits the logical error of assuming the converse, so it is the statement that is not necessarily true. \[ \boxed{\text{Option (C)}} \]
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