Question

Let \( A = \{1,2,3,\ldots,9\} \) and \( R \subset A \times A \) be defined by \[ (x,y) \in R \iff |x-y| \text{ is a multiple of } 3. \] Consider the following statements:
S\(_1\): Number of elements in \( R \) is 36.
S\(_2\): \( R \) is an equivalence relation.
Which of the following is correct?

• A. \( S_1 \) and \( S_2 \) both correct
• B. \( S_1 \) is correct, but \( S_2 \) is not correct
• C. \( S_2 \) is correct, but \( S_1 \) is not correct
• D. \( S_1 \) and \( S_2 \) both incorrect

Hint:

To test equivalence relations, always verify reflexive, symmetric, and transitive properties separately.

Answer 1

The Correct Option is D

To solve the problem, let's analyze the statements one by one:

1. Statement \( S_1 \): Number of elements in \( R \) is 36.

   Set \( A = \{1, 2, 3, \ldots, 9\} \) and relation \( R \) is defined by \((x,y) \in R\) if \(|x-y|\) is a multiple of 3.

   We check when \( |x - y| \equiv 0 \ (\mathrm{mod} \ 3) \). We can group the elements of \( A \) based on their equivalence class modulo 3:

   • Class 0 modulo 3: \(\{3, 6, 9\}\)
   • Class 1 modulo 3: \(\{1, 4, 7\}\)
   • Class 2 modulo 3: \(\{2, 5, 8\}\)

   Each class with 3 elements can form \( \binom{3}{2} \times 2 + 3 = 6 + 3 = 9 \) pairs (because every pair (a, b) is considered twice as (a, b) and (b, a), plus the reflexive pairs (a, a)).

   Thus, for all three classes: \(3 \times 9 = 27\).

   Therefore, Statement \( S_1 \) claiming 36 elements is incorrect.

2. Statement \( S_2 \): \( R \) is an equivalence relation.

   For \( R \) to be an equivalence relation, it must satisfy reflexivity, symmetry, and transitivity.

   • Reflexive: For any \( x \), \(|x-x|=0\), which is a multiple of 3, so reflexive property holds.
   • Symmetric: If \((x,y) \in R\), \(|x-y|\) is a multiple of 3, so \(|y-x|\) is the same, hence symmetric property holds.
   • Transitive: If \((x,y) \in R\) and \((y,z) \in R\), then \(|x-y|\) and \(|y-z|\) are multiples of 3. Thus, \(|x-z| = |x-y + y-z|\) is also a multiple of 3 (using properties of modulus), so transitive property holds.

   Since \( R \) is reflexive, symmetric, and transitive, \( R \) is indeed an equivalence relation. Therefore, Statement \( S_2 \) is correct.

In conclusion, the correct answer to the problem is: \( S_1 \) is incorrect while \( S_2 \) is correct. The given answer "both incorrect" seems mistaken. However, based on the problem setup, the outcome should logically see \( S_1 \) incorrect and \( S_2 \) correct.