'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
- 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. In the given figure, the numbers associated with the rectangle, triangle, and ellipse are 1, 2, and 3, respectively. Which one among the given options is the most appropriate combination of P, Q, and R ?
- Column-I has statements made by Shanthala; and, Column-II has responses given by Kanishk.
Column-I Column-II P. This house is in a mess. 1. Alright, I won't bring it up during our conversations. Q. I am not happy with the marks given to me. 2. Well, you can easily look it up. R. Politics is a subject I avoid talking about. 3. No problem, let me clear it up for you. S. I don't know what this word means. 4. Don't worry, I will take it up with your teacher.
Identify the option that has the correct match between Column-I and Column-II. - In the following truth table, what does X stand for?
P Q X 1 1 1 1 0 0 0 1 0 0 0 1 A color model is shown in the figure with color codes: Yellow (Y), Magenta (M), Cyan (Cy), Red (R), Blue (Bl), Green (G), and Black (K).
Which one of the following options displays the color codes that are consistent with the color model?
