JEE Mathematics 2026 Question with Solution

Given: $A = \{0,1,2,\dots,9\}$ and $(x,y) \in R$ iff $|x-y|$ is a multiple of $3$.

Find: Which of the two statements is correct.

The condition that $|x-y|$ is a multiple of $3$ means exactly that

So the relation groups elements according to their remainders when divided by $3$.

The equivalence classes are:

Since congruence modulo $3$ is reflexive, symmetric, and transitive, $R$ is an equivalence relation. Therefore, Statement II is correct.

Now count the ordered pairs in $R$. Every pair of elements from the same class belongs to $R$.

Substituting the class sizes:

So Statement I is incorrect because it claims $36$, whereas the actual value is $34$.

Therefore, the correct option is C.

Given: The relation is defined by $|x-y|$ being divisible by $3$.

Find: The truth values of Statement I and Statement II.

First rewrite the condition in modular form:

Hence two elements are related exactly when they have the same remainder modulo $3$.

Partition $A$ into residue classes modulo $3$:

• $C_0 = \{0,3,6,9\}$
• $C_1 = \{1,4,7\}$
• $C_2 = \{2,5,8\}$

A relation defined by equality of remainders modulo $3$ is an equivalence relation because:

• Reflexive: every element has the same remainder as itself.
• Symmetric: if $x$ and $y$ have the same remainder, then $y$ and $x$ also have the same remainder.
• Transitive: if $x$ and $y$ have the same remainder, and $y$ and $z$ have the same remainder, then $x$ and $z$ have the same remainder. So Statement II is true.

To count $n(R)$, count ordered pairs within each class.

• From $C_0$: $4^2 = 16$ pairs
• From $C_1$: $3^2 = 9$ pairs
• From $C_2$: $3^2 = 9$ pairs

Hence,

So Statement I is false.

Therefore, Statement I is incorrect but Statement II is correct, so the correct option is C.