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?

Correct Answer:

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-by-step Explanation:

To find which statement is necessarily true when the given observation is false, we need to mathematically negate the original statement.

Let us define the variables and predicates to express the statement in first-order logic:
Let $S$ be the set of all students in the class, where $\left|S\right|=100$.
Let $\mathrm{score}\left(x\right)$ represent the score of student $x$.
Let $P(x,y)$ be the predicate that the absolute difference between the scores of student $x$ and student $y$ is at most 10 marks:

$P(x,y):\left|\mathrm{score}\right(x)-\mathrm{score}(y\left)\right|\le 10$

The original statement says: "For each student, there exists another student in the class such that their scores are at most ten marks away."
Expressing this in formal logic:

$\forall x\in S,\exists y\in S(y\ne x\wedge P(x,y\left)\right)$

We are given that this statement is false. Therefore, its negation must be true. Let us find the negation:

$\neg [\forall x\in S,\exists y\in S(y\ne x\wedge P(x,y)\left)\right]$

Applying the rules of quantifier negation (where $\neg \forall$ becomes $\exists \neg$, and $\neg \exists$ becomes $\forall \neg$):

$\exists x\in S,\neg [\exists y\in S(y\ne x\wedge P(x,y)\left)\right]$

Simplifying further:

$\exists x\in S,\forall y\in S,\neg (y\ne x\wedge P(x,y\left)\right)$

Using De Morgan's laws, $\neg (A\wedge B)$ is logically equivalent to $A\Rightarrow \neg B$:

$\exists x\in S,\forall y\in S(y\ne x\Rightarrow \neg P(x,y\left)\right)$

Here, $\neg P(x,y)$ means the scores of $x$ and $y$ are more than 10 marks away:

$\neg P(x,y):\left|\mathrm{score}\right(x)-\mathrm{score}(y\left)\right|>10$

Translating this negated formula back into natural language:
"There exists at least one student ($\exists x\in S$) such that for all other students ($\forall y\in S$ where $y\ne x$), their score is more than 10 marks away ($\left|\mathrm{score}\right(x)-\mathrm{score}(y\left)\right|>10$)."

This matches the correct option exactly.