Let A = { 0 , 1 , 2 , … , 9 } Let R be a relation on A defined by ( x , y ) R if and only if ? x ? y ? is a multiple of 3. [2026]

Question

Let $A = {0, 1, 2, \dots, 9}$ 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. [2026]

Smart Achievers · Accepted Answer

(2)

$Number of form 3 K = 4$

$Number of form 3 K + 1 = 3$

$Number of form 3 K + 2 = 4$

$4 \times 4 + 3 \times 3 + 3 \times 3 = 34 relations$

$\Rightarrow x R y \Rightarrow y R x$

$\Rightarrow (x - y) = 3 λ, (y - z) = 3 μ$

$\Rightarrow (x - z) = 3 (λ + μ)$

$R is reflexive, symmetric and transitive S_{2} is true$

$Ans. S_{1} is false but S_{2} is true$

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