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

The relation corresponds to $x \equiv y \pmod 3$, which is a known equivalence relation (Reflexive, Symmetric, Transitive). Statement II is correct.
To find $n(R)$, we group elements by their remainder modulo 3:
$C_0 = \{0, 3, 6, 9\}$ (4 elements).
$C_1 = \{1, 4, 7\}$ (3 elements).
$C_2 = \{2, 5, 8\}$ (3 elements).
Pairs in R are formed by taking any two elements (including same) from the same group.
$n(R) = |C_0|^2 + |C_1|^2 + |C_2|^2$.
$n(R) = 4^2 + 3^2 + 3^2 = 16 + 9 + 9 = 34$.
Statement I says 36, so it is incorrect.