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

Step 1: Partition the set \( A \).
Elements of \( A \) can be grouped based on their remainder modulo 3: \[ \{1,4,7\},\quad \{2,5,8\},\quad \{3,6,9\} \]
Step 2: Count elements of relation \( R \).
Each group has 3 elements, giving \( 3 \times 3 = 9 \) ordered pairs per group.
Total elements in \( R = 3 \times 9 = 27 \).
Hence, \( S_1 \) is incorrect.
Step 3: Check equivalence relation properties.
Reflexive: \( |x-x| = 0 \), which is a multiple of 3
Symmetric: If \( |x-y| \) is a multiple of 3, then so is \( |y-x| \)
Transitive: Transitivity does not hold for all pairs
Step 4: Conclusion.
Since \( R \) is not transitive, it is not an equivalence relation.
Thus, both \( S_1 \) and \( S_2 \) are incorrect.