A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set A=\1, 2, 3\ : R_1=\(2,3),(3,2)\ R_2=\(1,2),(1,3),(3,2)\ R_3=\(1,2),(2,1),(1,1)\ R_4=\(1,1),(1,2),(3,3),(2,2)\ R_5=\(1,1),(1,2),(3,3),(2,2),(2,1),(2,3),(3,2)\ The students are asked to answer the following questions about the above relations : (i) Identify the relation which is reflexive, transitive but not symmetric. (ii) Identify the relation which is reflexive and symmetric but not transitive. (iii)(a) Identify the relations which are symmetric but neither reflexive nor transitive. (iii)(b) What pairs should be added to the relation R_2 to make it an equivalence relation ?

MathsCBSE Class 12 · Board exam 2025, set 65/1/1 · 4 marks · Case-based

CBSE Class 12 Mathematics 2025: question 37

Set in the CBSE Class 12 board exam of 2025, paper 65/1/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedRelations and Functions

A class-room teacher is keen to assess the learning of her students the concept of "relations" taught to them. She writes the following five relations each defined on the set $A = {1, 2, 3}$ :
$R_{1} = {(2, 3), (3, 2)}$
$R_{2} = {(1, 2), (1, 3), (3, 2)}$
$R_{3} = {(1, 2), (2, 1), (1, 1)}$
$R_{4} = {(1, 1), (1, 2), (3, 3), (2, 2)}$
$R_{5} = {(1, 1), (1, 2), (3, 3), (2, 2), (2, 1), (2, 3), (3, 2)}$
The students are asked to answer the following questions about the above relations :

(i)
Identify the relation which is reflexive, transitive but not symmetric.
1 mark
(ii)
Identify the relation which is reflexive and symmetric but not transitive.
1 mark
(iii)(a)
Identify the relations which are symmetric but neither reflexive nor transitive.
2 marks
OR
(iii)(b)
What pairs should be added to the relation $R_{2}$ to make it an equivalence relation ?
2 marks

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 65/1/1.

Hints

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

Hint 1

For each relation, check three things: does it contain (1,1), (2,2), (3,3)? Does every pair have its reverse? Do chains (a,b), (b,c) force (a,c)?

Hint 2

A pair such as (2,3), (3,2) requires (2,2) for transitivity.

Hint 3

For R₂, first add the missing diagonal pairs, then the reverses, and then check transitivity.

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.

(i) $R_{4}$ contains $(1, 1), (2, 2), (3, 3)$ , so it is reflexive. Every chain stays inside $R_{4}$ (e.g. $(1, 1), (1, 2) \to (1, 2)$ and $(1, 2), (2, 2) \to (1, 2)$ ), so it is transitive. But $(1, 2) \in R_{4}$ while $(2, 1) \in / R_{4}$ , so it is not symmetric. Answer: $R_{4}$ .
1 mark
(ii) $R_{5}$ contains all of $(1, 1), (2, 2), (3, 3)$ , so it is reflexive. Every pair in it has its reverse, so it is symmetric. But $(1, 2), (2, 3) \in R_{5}$ while $(1, 3) \in / R_{5}$ , so it is not transitive. Answer: $R_{5}$ .
1 mark
(iii)(a) $R_{1}$ is symmetric. It is not reflexive since $(1, 1) \in / R_{1}$ . It is not transitive, because $(2, 3), (3, 2) \in R_{1}$ but $(2, 2) \in / R_{1}$ .
1 mark
(iii)(a) $R_{3}$ is symmetric. It is not reflexive since $(2, 2) \in / R_{3}$ . It is not transitive, because $(2, 1), (1, 2) \in R_{3}$ but $(2, 2) \in / R_{3}$ . So the required relations are $R_{1}$ and $R_{3}$ .
1 mark
(iii)(b) For reflexivity, add $(1, 1), (2, 2), (3, 3)$ . For symmetry, add the reverses $(2, 1), (3, 1), (2, 3)$ .
1 mark
(iii)(b) The relation then becomes $A \times A$ , which is transitive and hence an equivalence relation. The pairs to add are $(1, 1), (2, 2), (3, 3), (2, 1), (3, 1), (2, 3)$ .
1 mark

Answer: (i) $R_{4}$ ; (ii) $R_{5}$ ; (iii)(a) $R_{1}$ and $R_{3}$ ; (iii)(b) add $(1, 1), (2, 2), (3, 3), (2, 1), (3, 1), (2, 3)$

Where the marks are

(i) Identifies $R_{4}$ (1 mark)
(ii) Identifies $R_{5}$ (1 mark)
(iii)(a) Identifies $R_{1}$ (1 mark) and $R_{3}$ (1 mark)
(iii)(b) Lists all six pairs to be added (2 marks)

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 paper2025 board paper, set 65/1/1: every question, by section.

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CBSE Class 12 Mathematics 2025: question 37

Hints

Step-by-step solution

Revise the chapter

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