A relation R is defined on Z, the set of integers, as R = \(x, y) : x - y is divisible by a prime number 'p', x, y Z\ check whether R is an equivalence relation or not.

MathsCBSE Class 12 · Board exam 2026, set 65/1/1 · 5 marks · Long answer

CBSE Class 12 Mathematics 2026: question 32(a)

Set in the CBSE Class 12 board exam of 2026, paper 65/1/1, section D, question 32(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q32(a) · 5 marks · Long answerRelations and Functions

A relation R is defined on Z, the set of integers, as
$R = {(x, y) : ∣ x - y ∣$ is divisible by a prime number 'p', $x, y \in Z}$
check whether R is an equivalence relation or not.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 65/1/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

Check the three properties one by one.

Hint 2

Remember 0 is divisible by every prime.

Hint 3

For transitivity, write $x - y = k p$ , $y - z = m p$ and add.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Reflexive: For any $x \in Z$ , $∣ x - x ∣ = 0$ , which is divisible by $p$ ( $0 = 0 \cdot p$ ). So $(x, x) \in R$ and R is reflexive.
1½ marks
Symmetric: Let $(x, y) \in R$ , so $∣ x - y ∣$ is divisible by $p$ . Since $∣ y - x ∣ = ∣ x - y ∣$ , $∣ y - x ∣$ is divisible by $p$ , so $(y, x) \in R$ . R is symmetric.
1½ marks
Transitive: Let $(x, y), (y, z) \in R$ . Then $x - y = k p$ and $y - z = m p$ for some integers $k, m$ . Adding, $x - z = (k + m) p$ , so $∣ x - z ∣$ is divisible by $p$ and $(x, z) \in R$ . R is transitive.
1½ marks
Since R is reflexive, symmetric and transitive, R is an equivalence relation.
½ mark

Answer: R is an equivalence relation

Where the marks are

Proves reflexivity using $∣ x - x ∣ = 0$ (1½ marks)
Proves symmetry using $∣ x - y ∣ = ∣ y - x ∣$ (1½ marks)
Proves transitivity by adding $x - y$ and $y - z$ (1½ marks)
Concludes equivalence relation (½ mark)

Revise the chapter

Chapter: Relations and Functions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Relations and Functions, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 65/1/1: every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 12 Mathematics 2026: question 32(a)

Hints

Step-by-step solution

Revise the chapter

More questions from Relations and Functions