'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?
Exam Scores Logic: Understanding the Statement
The original statement asserts: "For each student, there exists another student whose score is at most 10 marks away." This implies that no student is completely isolated score-wise from everyone else by more than 10 marks.
Negating the Exam Scores Statement
We are given that the original statement is false. To determine what must be true, we find the logical negation of the statement.
Original Statement (Symbolic): $\forall \text{ student } S_i, \exists \text{ student } S_j (j \neq i) \text{ such that } |Score(S_i) - Score(S_j)| \le 10$.
The negation of this statement is:
$\neg (\forall S_i, \exists S_j (j \neq i) \text{ such that } |Score(S_i) - Score(S_j)| \le 10)$
Applying rules of logical negation (negating quantifiers and the condition):
$\exists \text{ student } S_k \text{ such that } \neg (\exists S_j (j \neq k) \text{ such that } |Score(S_k) - Score(S_j)| \le 10)$
This further simplifies to:
$\exists \text{ student } S_k \text{ such that } \forall \text{ student } S_j (j \neq k), \neg (|Score(S_k) - Score(S_j)| \le 10)$
Final Negated Statement: $\exists \text{ student } S_k \text{ such that } \forall \text{ student } S_j (j \neq k), |Score(S_k) - Score(S_j)| > 10$.
In words: "There exists at least one student ($S_k$) for whom all other students ($S_j$) have scores more than 10 marks away."
Identifying the Necessarily True Statement
We now compare the derived negation with the given options:
- Option 1 is too strong, requiring the condition for *all* students, not just one.
- Option 2 matches the derived negation exactly: "There exists at least one student in the class for whom the scores of all the other students are more than 10 marks away."
- Options 3 and 4 introduce specific counts ("exactly one") which are not guaranteed by the logical negation.
Thus, Option 2 is the only statement that is necessarily true.
Important Questions from Logical Deduction
- Consider the following two phonological rules:
Rule 1: Vowel Epenthesis of [ɪ] after sibilant-ending stems
Rule 2: Progressive Voicing Assimilation of the plural affix
Which ONE of the following rule ordering relations apply in the case of regular English pluralization as in ‘horse’ – ‘horses’? - Based only on the conversation below, identify the logically correct inference:
“Even if I had known that you were in the hospital, I would not have gone there to see you", Ramya told Josephine. Consider a five-digit number PQRST that has distinct digits P, Q, R, S and T, and satisfies the following conditions:
$P < Q$
$S > P > T$
$R < T$
If integers 1 through 5 are used to construct such a number, the value of P is:Residency is a famous housing complex with many well-established individuals among its residents. A recent survey conducted among the residents of the complex revealed that all of those residents who are well established in their respective fields happen to be academicians. The survey also revealed that most of these academicians are authors of some best-selling books.
Based only on the information provided above, which one of the following statements can be logically inferred with certainty?- The police arrested four criminals – P, Q, R and S. The criminals knew each other. They made the following statements:
P says "Q committed the crime.”
Q says "S committed the crime. "
R says"I did not do it."
S says "What Q said about me is false."
Assume only one of the arrested four committed the crime and only one of the statements made above is true. Who committed the crime?
