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Short Intro

In this post, students can find complete step-by-step solutions for Class 12 Maths Chapter 1 Exercise 1.1 – Relations and Functions based on the latest NCERT syllabus. This exercise covers reflexive, symmetric, transitive, and equivalence relations in a simple and exam-oriented format.

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Quick Information Box

Item	Details
Board	NCERT / CBSE
Class	12
Subject	Mathematics
Chapter	Relations and Functions
Exercise	1.1
Main Topics	Relations & Equivalence Relations

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Concepts Used (Topics Covered)

• Reflexive Relation
• Symmetric Relation
• Transitive Relation
• Equivalence Relation
• Universal Relation
• Empty Relation
• Equivalence Classes

The chapter explains different types of relations and equivalence relations.

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Important Formulas

Reflexive Relation

(a, a) ∈ R for every a ∈ A

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Symmetric Relation

(a, b) ∈ R ⇒ (b, a) ∈ R

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Transitive Relation

(a, b) ∈ R and (b, c) ∈ R ⇒ (a, c) ∈ R

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Equivalence Relation

A relation which is reflexive, symmetric and transitive.

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Questions & Step-by-step Solutions

Question 1

Determine whether the following relations are reflexive, symmetric and transitive.

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(i) Relation R in A = {1, 2, 3, …, 14} defined by:

R = {(x, y) : 3x − y = 0}

Solution

Given:

y = 3x

Checking reflexive:

For reflexive relation:

3x − x = 0
⇒ 2x = 0

Not true for all x.

Therefore:

R is not reflexive.

Checking symmetric:

If:

3x − y = 0
⇒ y = 3x

Then:

3y − x ≠ 0

in general.

Therefore:

R is not symmetric.

Checking transitive:

If:

y = 3x and z = 3y

Then:

z = 9x

But relation requires:

z = 3x

Not true.

Therefore:

R is not transitive.

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(ii) Relation in N:

R = {(x, y) : y = x + 5 and x < 4}

Solution

Possible ordered pairs:

(1,6), (2,7), (3,8)

No pair of form:

(x, x)

Hence:

Not reflexive.

Also:

(1,6) ∈ R but (6,1) ∉ R

Hence:

Not symmetric.

No chain exists for transitivity.

Therefore:

R is not transitive.

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(iii) Relation in A = {1,2,3,4,5,6}

R = {(x, y) : y is divisible by x}

Solution

Every number divides itself.

Therefore:

R is reflexive.

But:

2 divides 4

while:

4 does not divide 2

Therefore:

R is not symmetric.

If:

x divides y and y divides z

then:

x divides z

Therefore:

R is transitive.

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(iv) Relation in Z:

R = {(x, y) : x − y is an integer}

Solution

Since x and y are integers:

x − y is always an integer

Hence relation is:

Reflexive, symmetric and transitive.

Therefore:

R is an equivalence relation.

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(v) Relations among human beings

(a) Work at same place

Reflexive, symmetric and transitive

Hence equivalence relation.

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(b) Live in same locality

Reflexive, symmetric and transitive

Hence equivalence relation.

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(c) x is exactly 7 cm taller than y

Neither reflexive nor symmetric nor transitive

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(d) x is wife of y

Neither reflexive nor transitive

Not symmetric.

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(e) x is father of y

Neither reflexive nor symmetric nor transitive

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Question 2

Show that relation:

R = {(a,b) : a ≤ b²}

is neither reflexive nor symmetric nor transitive.

Solution

Reflexive:

For all a:

a ≤ a²

fails for:

a = 1/2

Hence not reflexive.

Symmetric:

Take:

a = 2, b = 3

Then:

2 ≤ 9

true.

But:

3 ≤ 4

may fail for other values.

Hence not symmetric.

Transitive also fails.

Therefore:

R is neither reflexive nor symmetric nor transitive.

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Question 3

Relation in {1,2,3,4,5,6}

R = {(a,b): b = a + 1}

Solution

Pairs:

(1,2), (2,3), (3,4), (4,5), (5,6)

No pair:

(a,a)

Hence not reflexive.

Also:

(1,2) ∈ R but (2,1) ∉ R

Hence not symmetric.

Further:

(1,2), (2,3) ∈ R

but:

(1,3) ∉ R

Hence not transitive.

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Question 4

Show that:

R = {(a,b): a ≤ b}

is reflexive and transitive but not symmetric.

Solution

Since:

a ≤ a

R is reflexive.

If:

a ≤ b and b ≤ c

then:

a ≤ c

Hence transitive.

But:

2 ≤ 5

does not imply:

5 ≤ 2

Hence not symmetric.

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Question 5

Check relation:

R = {(a,b): a ≤ b³}

Solution

Reflexive:

a ≤ a³

not always true.

Hence not reflexive.

Symmetric and transitive also fail.

Therefore:

R is neither reflexive nor symmetric nor transitive.

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Common Mistakes

• Confusing symmetric with transitive relation
• Forgetting ordered pair direction
• Missing counterexamples
• Incorrectly assuming every relation is reflexive

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Exam Tips

• Always test all three properties separately.
• Use counterexamples to disprove properties.
• Write proper mathematical statements.
• Learn definitions thoroughly.

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Practice MCQs

MCQ 1

A relation which is reflexive, symmetric and transitive is called:

A. Universal relation
B. Empty relation
C. Equivalence relation
D. Identity relation

Answer:

C. Equivalence relation

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MCQ 2

If (a,b) ∈ R implies (b,a) ∈ R, then relation is:

A. Reflexive
B. Symmetric
C. Transitive
D. Universal

Answer:

B. Symmetric

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MCQ 3

Relation:

a ≤ b

is:

A. Symmetric
B. Reflexive only
C. Reflexive and transitive
D. None

Answer:

C. Reflexive and transitive

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FAQ Section

What is a reflexive relation?

A relation where:

(a,a) ∈ R for every a ∈ A

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What is symmetric relation?

If:

(a,b) ∈ R ⇒ (b,a) ∈ R

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What is transitive relation?

If:

(a,b) ∈ R and (b,c) ∈ R

then:

(a,c) ∈ R

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What is equivalence relation?

A relation which is reflexive, symmetric and transitive.

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