'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
Compare each option to the original statement's contrapositive, its weaker consequences, and its converse. Only the converse is not guaranteed.
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.
Show Solution
Verified By Collegedunia
The Correct Option isC
Solution and Explanation
The given statement has the logical form "If the teacher is in the room, then all students stand silently," written as \(T \rightarrow S\), where \(T\) is "the teacher is in the room" and \(S\) is "all students stand silently."
Option (A) states "If any student is not standing silently, then the teacher is not in the room," which is exactly the contrapositive of \(T \rightarrow S\), namely "not \(S\) implies not \(T\)." A statement and its contrapositive are always logically equivalent, so this must necessarily be true.
Option (B) states "When the teacher is in the room, all students are silent." Being silent is a weaker requirement than standing silently, so \(T \rightarrow S\) directly guarantees this weaker consequence. This is necessarily true.
Option (D) states "When the teacher is in the room, all students are standing." Standing is likewise a weaker requirement than standing silently, so this too follows directly from the given statement and is necessarily true.
Option (C) states "If all students are standing, then the teacher is in the room." This reverses the direction of the original implication, trying to conclude the teacher's presence from students standing, but the original statement only tells us what happens when the teacher is present. It says nothing about why students might be standing at some other time, for example during a fire drill, an assembly, or by their own choice while the teacher is elsewhere. So students standing does not guarantee that the teacher is in the room.
Since option (C) attempts to conclude the antecedent from the consequent without justification, it is the one statement that is not necessarily true.
