‘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?
‘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?
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
Step 2: A conditional \(p \rightarrow q\) is logically equivalent to its contrapositive \(\lnot q \rightarrow \lnot p\): "If any student is not standing silently, then the teacher is not in the room." This is Option A, and it is necessarily true.
Step 3: "Standing silently" logically implies both "standing" and "silent" separately. So \(p \rightarrow q\) also guarantees \(p \rightarrow \text{silent}\) (Option B) and \(p \rightarrow \text{standing}\) (Option D) - both are necessarily true as weaker consequences of \(q\).
Step 4: Option C claims the converse-like statement "If all students are standing, then the teacher is in the room," i.e., \(\text{standing} \rightarrow p\). A conditional never implies its converse - students could be standing for an unrelated reason (fire drill, announcement) even without the teacher present. So this is NOT necessarily true.
Final Answer: \(\boxed{\text{Option C}}\)
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