JEE Main 2026 — Jan 23, Shift 2 — Mathematics solution

JEE Main2026Jan 23, Shift 2Mathematics

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

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

Solution

Partition $A$ by residue modulo $3$ :

${0, 3, 6, 9}$ has 4 elements; ${1, 4, 7}$ has 3; ${2, 5, 8}$ has 3.

$(x, y) \in R$ iff $x$ and $y$ lie in the same residue class. So

$n (R) = 4^{2} + 3^{2} + 3^{2} = 16 + 9 + 9 = 34 \neq = 36$ . Statement I is incorrect.

For Statement II: $R$ is reflexive ( $∣ x - x ∣ = 0$ , divisible by 3), symmetric ( $∣ x - y ∣ = ∣ y - x ∣$ ), and transitive (if $3 ∣ (x - y)$ and $3 ∣ (y - z)$ , then $3 ∣ (x - z)$ ). So $R$ is an equivalence relation. Statement II is correct.

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