Question

The following observation is made about the scores obtained by 100 students in an exam:
'For each student, there exists another student in the class such that their scores are at most ten marks away.'
If the above statement is false, which one of the following statements is necessarily true?

• A. For each student, the scores of all the other students are more than 10 marks away.
• B. There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away.
• C. There is exactly one student in the class for whom the scores of some students are more than 10 marks away.
• D. For each student, the score of exactly one other student is more than 10 marks away.

Hint:

Negating "for every student there is a close one" flips the quantifiers: it becomes "some student has no close one".

Answer 1

The Correct Option is B

Step 1: Write the given statement using quantifiers.
Let the students be labelled \(1\) to \(100\), and let \(s(i)\) be the score of student \(i\). The given statement says:
"For every student \(i\), there exists some other student \(j\) such that \(|s(i)-s(j)|\le 10\)."
In symbols, this is: for all \(i\), there exists \(j\neq i\) with \(|s(i)-s(j)|\le 10\).

Step 2: Recall how to negate a "for all, there exists" statement.
The negation of "for all \(i\), there exists \(j\) such that \(P(i,j)\)" is "there exists an \(i\) such that for all \(j\), \(P(i,j)\) is false." The "for all" flips to "there exists", the "there exists" flips to "for all", and the inner condition gets negated.

Step 3: Apply this to our statement.
The inner condition here is \(|s(i)-s(j)|\le 10\), whose negation is \(|s(i)-s(j)|>10\).
So the negation of the whole statement is: "there exists a student \(i\) such that for every other student \(j\), \(|s(i)-s(j)|>10\)."
In plain words: there is at least one student whose score is more than \(10\) marks away from every other student's score.

Step 4: Match this to the options.
This is exactly what option (B) says: "There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away."

Step 5: Check why option (A) is wrong.
Option (A) says this holds for EVERY student, not just for at least one. Negating a "there exists" statement only gives us one such student, not all \(100\) of them.

Step 6: Check why option (C) is wrong.
Option (C) restricts the count to "exactly one" student and also weakens the condition to "some students" instead of "all other students". Neither restriction is forced by the negation.

Step 7: Check why option (D) is wrong.
Option (D) says "for each student, exactly one other student is more than 10 marks away," which is a completely different statement about counting distances for every student, not about the existence of one isolated student. It is not what the negation produces.

Step 8: Final conclusion.
\[ \boxed{\text{There exists at least one student for whom the scores of all other students are more than 10 marks away.}} \]
Hence the correct option is (B).