Question

Let \( A = \{0,1,2,\ldots,9\} \). Let \( R \) be a relation on \( A \) defined by \((x,y) \in R\) if and only if \( |x - y| \) is a multiple of \(3\). Given below are two statements:
Statement I: \( n(R) = 36 \).
Statement II: \( R \) is an equivalence relation.
In the light of the above statements, choose the correct answer from the options given below.

• A. Both Statement I and Statement II are correct
• B. Both Statement I and Statement II are incorrect
• C. Statement I is incorrect but Statement II is correct
• D. Statement I is correct but Statement II is incorrect

Hint:

For an equivalence relation, the graph is a union of disjoint squares corresponding to the equivalence classes.

Answer 1

The Correct Option is C

To solve the given problem, we need to analyze both statements regarding the relation \( R \) on the set \( A = \{0, 1, 2, \ldots, 9\} \).

1. Statement I: \( n(R) = 36 \)
   • The set \( A \) consists of integers from 0 to 9. The relation \( R \) is defined such that \((x, y) \in R\) if and only if \(|x - y|\) is a multiple of 3. 
   • Let's calculate the pairs \((x, y)\) such that \(|x - y|\) is a multiple of 3.
   • The multiples of 3 are \(\{0, 3, 6, 9\}\). Thus, for each element \( x \) in \( A \), \( y \) can be the set made by adding each of these multiples of 3 to \( x \), ensuring \( y \) remains within \( \{0, 1, 2, \ldots, 9\} \).
   • Examine \( x = 0 \) as an example:
     • \(|0 - 0|\), \(|0 - 3|\), \(|0 - 6|\), \(|0 - 9|\) are multiples of 3.
     • Therefore, valid \( y \) values are 0, 3, 6, and 9.
   • Similarly, for any \( x \), possible \( y \) values are those with the same residue modulo 3. Hence, there are 4 such values for each \( x \).
   • Since the set \( A \) is divided based on residues modulo 3 (0, 1, 2), each class has 3 numbers (e.g., \(\{0, 3, 6, 9\}\), \(\{1, 4, 7\}\), \(\{2, 5, 8\}\)). Each number in one class pairs with all numbers within their class.
   • Therefore, total number of such pairs \( = \underbrace{4}_{(0,3,6,9)} \times \underbrace{4}_{(times the same set for each residue)} + \underbrace{3}_{(1,4,7)} \times \underbrace{3} + \underbrace{3}_{(2,5,8)} \times \underbrace{3} = 16 + 9 + 9 = 34\).
   • Thus, \( n(R) = 34 \), not 36. Therefore, Statement I is incorrect.
2. Statement II: \( R \) is an equivalence relation
   • A relation is an equivalence relation if it is reflexive, symmetric, and transitive.
   • Reflexive: For all \( x \in A \), \(|x - x| = 0\) which is a multiple of 3. Hence it is reflexive.
   • Symmetric: If \(|x - y|\) is a multiple of 3, then \(|y - x| = |x - y|\) is also a multiple of 3, making it symmetric.
   • Transitive: Assume \((x, y) \in R\) and \((y, z) \in R\), i.e., \(|x - y|\) and \(|y - z|\) are multiples of 3. By addition, \(|x - z| = |x - y| + |y - z|\) is a multiple of 3, so the relation is transitive.
   • Therefore, \( R \) satisfies all conditions of an equivalence relation. Statement II is correct.

In conclusion, Statement I is incorrect but Statement II is correct.