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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

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(ii) Relation R in the set N of natural numbers defined as

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

Solution

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

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

Solution

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(iv) Relation R in the set Z of all integers defined as

R = {(x, y) : x − y is an integer}

Solution

(v) Relation R in the set A of human beings in a town at a particular time given by

(a) R = {(x, y) : x and y work at the same place}
(b) R = {(x, y) : x and y live in the same locality}
(c) R = {(x, y) : x is exactly 7 cm taller than y}
(d) R = {(x, y) : x is wife of y}
(e) R = {(x, y) : x is father of y}

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Question 2

Show that the relation R in the set R of real numbers, defined as
R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.

Solution

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Question 3

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric or transitive.

Solution

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Question 4

Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive and transitive but not symmetric.

Solution

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Question 5

Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive,
symmetric or transitive.

Solution

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Question 6

Show that the relation R in the set {1, 2, 3} given by R = {(1, 2), (2, 1)} is
symmetric but neither reflexive nor transitive.

Solution

Question 7

Show that the relation R in the set A of all the books in a library of a college,
given by R = {(x, y) : x and y have same number of pages} is an equivalence
relation.

Solution

Question 8

Show that the relation R in the set A = {1, 2, 3, 4, 5} given by
R = {(a, b) : |a – b| is even}, is an equivalence relation. Show that all the
elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are
related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.

Solution

Question 9

Show that each of the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by
(i) R = {(a, b) : |a – b| is a multiple of 4}
(ii) R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.

Solution

Question 10

Give an example of a relation. Which is
(i) Symmetric but neither reflexive nor transitive.
(ii) Transitive but neither reflexive nor symmetric.
(iii) Reflexive and symmetric but not transitive.
(iv) Reflexive and transitive but not symmetric.
(v) Symmetric and transitive but not reflexive.

Solution

Question 11

Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

Solution

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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