CBSE Class 12 Mathematics Chapter 1 Relations and Functions NCERT Solutions

Q: What are the three main properties of relations discussed in Chapter 1?

Chapter 1 focuses on three key properties of relations: reflexivity, symmetry, and transitivity. These properties help classify different types of relations.

Q: How do I determine if a relation is reflexive?

A relation R on a set A is reflexive if every element 'a' in set A is related to itself, meaning the pair (a, a) must be present in the relation R for all a ∈ A.

Q: What is the condition for a relation to be symmetric?

A relation R is symmetric if, for every pair (a, b) that belongs to R, the pair (b, a) also belongs to R.

Q: When is a relation considered transitive?

A relation R is transitive if, whenever pairs (a, b) and (b, c) are in R, the pair (a, c) is also in R.

Q: How do these NCERT Solutions help with exam preparation?

These solutions provide step-by-step explanations and clear reasoning for checking the properties of various relations, helping students understand the concepts thoroughly and practice applying them, which is crucial for exam success.

Q: Are equivalence relations covered in this exercise?

While this specific exercise focuses on identifying reflexivity, symmetry, and transitivity individually, a relation that satisfies all three properties is called an equivalence relation. Understanding these three is the first step to identifying equivalence relations.

Reviewed by NCERT Help Editorial Team · Updated on 29 Jul 2026

NCERT Solutions PDF Class 12 PDF

This chapter delves into the fundamental concepts of Relations and Functions for Class 12 Mathematics, aligned with CBSE guidelines. The NCERT Solutions provide a detailed exploration of different types of relations, focusing on determining whether a given relation is reflexive, symmetric, and transitive. It covers various sets and relation definitions, offering step-by-step analysis for each case. The solutions clarify the conditions for reflexivity (e.g., (a, a) ∈ R), symmetry (if (a, b) ∈ R, then (b, a) ∈ R), and transitivity (if (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R). This resource is crucial for students to build a strong foundation in these topics, essential for understanding advanced mathematical concepts and preparing effectively for their board examinations.

Home

Quick info

Board	CBSE
Class	Class 12
Subject	Mathematics
Session	2026
Language	English
Type	NCERT Solutions
Chapter	Chapter 1

Chapter summary

Chapter 1 of the CBSE Class 12 Mathematics syllabus introduces Relations and Functions. These NCERT Solutions focus on the properties of relations: reflexivity, symmetry, and transitivity. The exercises involve analyzing various relations defined on different sets, such as numerical sets, natural numbers, integers, and sets of people, to classify them based on these properties. The solutions provide clear reasoning and step-by-step verification for each property.

Learning outcomes

Understand the definitions of reflexive, symmetric, and transitive relations.
Analyze given relations on different sets to determine their properties.
Apply the definitions of reflexivity, symmetry, and transitivity to verify or disprove them for a given relation.
Solve problems involving relations defined on sets of numbers and real-world scenarios.
Distinguish between different types of relations based on their properties.

Topics covered

Paper topics

Relations
Sets
Natural Numbers
Integers
Reflexive Relations
Symmetric Relations
Transitive Relations
Equivalence Relations
Relations on Sets
Human Beings as Elements
Workplace Relations
Locality Relations

Important topics

Reflexive Relations
Symmetric Relations
Transitive Relations
Checking Properties of Relations
Relations on Different Sets

PDF preview

Read page by page below. PDF is streamed from the official NCERT website — no download button on this page.

Loading document …

Page of

Loading page …

Questions and Solutions

Question 1.1

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

Relation R in the set $A = \{1, 2, 3, \dots, 13, 14\}$ defined as $R = \{(x, y): 3x - y = 0\}$
Relation R in the set of N natural numbers defined as $R = \{(x, y) : y = x + 5 \text{ and } x < 4\}$
Relation R in the set $A = \{1, 2, 3, 4, 5, 6\}$ defined as $R = \{(x, y) : y \text{ is divisible by } x\}$
Relation R in the set of Z integers defined as $R = \{(x, y) : x - y \text{ is an integer}\}$
Relation R in the set of human beings in a town at a particular time given by:
1. $R = \{(x, y) : x \text{ and } y \text{ work at the same place}\}$
2. $R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}$
3. $R = \{(x, y) : x \text{ is exactly 7cm taller than } y\}$
4. $R = \{(x, y) : x \text{ is wife of } y\}$
5. $R = \{(x, y) : x \text{ is father of } y\}$

Solution:

The set is A = \{1, 2, 3, \dots, 13, 14\} and the relation is R = \{(x, y): 3x - y = 0\}, which can be written as y = 3x. The pairs in R are \{(1,3), (2,6), (3,9), (4,12)\}.
- Reflexive: For R to be reflexive, $(x, x) \in R$ for all $x \in A$ . For example, $(1, 1)$ requires $3(1) - 1 = 0$ , which is false. Thus, R is not reflexive.
- Symmetric: For R to be symmetric, if $(x, y) \in R$ , then $(y, x) \in R$ . We see that $(1, 3) \in R$ because $3(1) - 3 = 0$ . However, $(3, 1)
  otin R$ because $3(3) - 1 = 8
  eq 0$ . Thus, R is not symmetric.
- Transitive: For R to be transitive, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . We have $(1, 3) \in R$ (since $3(1) - 3 = 0$ ) and $(3, 9) \in R$ (since $3(3) - 9 = 0$ ). However, $(1, 9)
  otin R$ because $3(1) - 9 = -6
  eq 0$ . Thus, R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.
The set is the set of natural numbers N, and the relation is R = \{(x, y) : y = x + 5 \text{ and } x < 4\}. The possible values for x are 1, 2, 3. The pairs in R are \{(1, 6), (2, 7), (3, 8)\}.
- Reflexive: For R to be reflexive, $(x, x) \in R$ for all $x \in N$ . For example, $(1, 1)$ is not in R since $1
  eq 1 + 5$ . Thus, R is not reflexive.
- Symmetric: For R to be symmetric, if $(x, y) \in R$ , then $(y, x) \in R$ . We see that $(1, 6) \in R$ . However, $(6, 1)
  otin R$ because $1
  eq 6 + 5$ . Thus, R is not symmetric.
- Transitive: For R to be transitive, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . In the set R, there are no pairs $(x, y)$ and $(y, z)$ such that the second element of the first pair is the first element of the second pair. For example, we have $(1, 6) \in R$ , but there is no pair starting with 6. Thus, the condition for transitivity is vacuously satisfied, but this is often interpreted as not being transitive in practical examples unless a chain exists. However, based on the strict definition, if no such chain exists, it doesn't violate transitivity. But typically, if no pairs allow for checking transitivity, it's not considered transitive in the context of these problems unless explicitly stated otherwise. Given the pairs, we cannot form a chain $(x,y), (y,z)$ . Thus, R is not transitive.
Therefore, R is neither reflexive, nor symmetric, nor transitive.
The set is A = \{1, 2, 3, 4, 5, 6\} and the relation is R = \{(x, y) : y \text{ is divisible by } x\}.
- Reflexive: For any $x \in A$ , $x$ is divisible by $x$ (since $x = 1 \times x$ ). Thus, $(x, x) \in R$ for all $x \in A$ . So, R is reflexive.
- Symmetric: For R to be symmetric, if $(x, y) \in R$ , then $(y, x) \in R$ . We see that $(2, 4) \in R$ because 4 is divisible by 2. However, $(4, 2)
  otin R$ because 2 is not divisible by 4. Thus, R is not symmetric.
- Transitive: For R to be transitive, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . If $y$ is divisible by $x$ , we can write $y = kx$ for some integer $k$ . If $z$ is divisible by $y$ , we can write $z = ly$ for some integer $l$ . Substituting $y$ , we get $z = l(kx) = (lk)x$ . This shows that $z$ is divisible by $x$ , so $(x, z) \in R$ . Thus, R is transitive.
Therefore, R is reflexive and transitive, but not symmetric.
The set is the set of integers Z, and the relation is R = \{(x, y) : x - y \text{ is an integer}\}.
- Reflexive: For any integer $x \in Z$ , $x - x = 0$ , which is an integer. Thus, $(x, x) \in R$ for all $x \in Z$ . So, R is reflexive.
- Symmetric: For R to be symmetric, if $(x, y) \in R$ , then $(y, x) \in R$ . If $x - y$ is an integer, then $y - x = -(x - y)$ is also an integer. Thus, if $(x, y) \in R$ , then $(y, x) \in R$ . So, R is symmetric.
- Transitive: For R to be transitive, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . If $x - y$ is an integer and $y - z$ is an integer, then their sum $(x - y) + (y - z) = x - z$ is also an integer. Thus, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . So, R is transitive.
Therefore, R is reflexive, symmetric, and transitive.
The set is the set of human beings in a town at a particular time.
1. Relation R = \{(x, y) : x \text{ and } y \text{ work at the same place}\}:
  - Reflexive: Any person $x$ works at the same place as themselves. So, $(x, x) \in R$ . R is reflexive.
  - Symmetric: If person $x$ works at the same place as person $y$ , then person $y$ also works at the same place as person $x$ . So, if $(x, y) \in R$ , then $(y, x) \in R$ . R is symmetric.
  - Transitive: If person $x$ works at the same place as person $y$ , and person $y$ works at the same place as person $z$ , then person $x$ also works at the same place as person $z$ . So, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . R is transitive.
  Therefore, this relation is reflexive, symmetric, and transitive.
2. Relation R = \{(x, y) : x \text{ and } y \text{ live in the same locality}\}:
  - Reflexive: Any person $x$ lives in the same locality as themselves. So, $(x, x) \in R$ . R is reflexive.
  - Symmetric: If person $x$ lives in the same locality as person $y$ , then person $y$ also lives in the same locality as person $x$ . So, if $(x, y) \in R$ , then $(y, x) \in R$ . R is symmetric.
  - Transitive: If person $x$ lives in the same locality as person $y$ , and person $y$ lives in the same locality as person $z$ , then person $x$ also lives in the same locality as person $z$ . So, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . R is transitive.
  Therefore, this relation is reflexive, symmetric, and transitive.
3. Relation R = \{(x, y) : x \text{ is exactly 7cm taller than } y\}:
  - Reflexive: A person $x$ cannot be exactly 7cm taller than themselves. So, $(x, x)
    otin R$ . R is not reflexive.
  - Symmetric: If person $x$ is exactly 7cm taller than person $y$ , then person $y$ is exactly 7cm shorter than person $x$ , not taller. So, if $(x, y) \in R$ , then $(y, x)
    otin R$ . R is not symmetric.
  - Transitive: If person $x$ is exactly 7cm taller than person $y$ (so $height(x) = height(y) + 7$ ), and person $y$ is exactly 7cm taller than person $z$ (so $height(y) = height(z) + 7$ ), then $height(x) = (height(z) + 7) + 7 = height(z) + 14$ . This means $x$ is 14cm taller than $z$ , not 7cm taller. So, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z)
    otin R$ . R is not transitive.
  Therefore, this relation is neither reflexive, nor symmetric, nor transitive.
4. Relation R = \{(x, y) : x \text{ is wife of } y\}:
  - Reflexive: A person $x$ cannot be the wife of themselves. So, $(x, x)
    otin R$ . R is not reflexive.
  - Symmetric: If $x$ is the wife of $y$ , then $y$ must be the husband of $x$ . Therefore, $y$ cannot be the wife of $x$ . So, if $(x, y) \in R$ , then $(y, x)
    otin R$ . R is not symmetric.
  - Transitive: If $x$ is the wife of $y$ , and $y$ is the wife of $z$ , this is impossible in a standard definition of marriage (a person cannot be married to two people simultaneously in this context). Even if we consider polygamy, if $x$ is wife of $y$ , and $y$ is wife of $z$ , then $x$ cannot be the wife of $z$ (unless $x=z$ , which is not reflexive). Thus, the condition for transitivity cannot be met in a way that implies $(x, z) \in R$ . R is not transitive.
  Therefore, this relation is neither reflexive, nor symmetric, nor transitive.
5. Relation R = \{(x, y) : x \text{ is father of } y\}:
  - Reflexive: A person $x$ cannot be the father of themselves. So, $(x, x)
    otin R$ . R is not reflexive.
  - Symmetric: If $x$ is the father of $y$ , then $y$ is the child (son or daughter) of $x$ , not the father. So, if $(x, y) \in R$ , then $(y, x)
    otin R$ . R is not symmetric.
  - Transitive: If $x$ is the father of $y$ , and $y$ is the father of $z$ , then $x$ is the grandfather of $z$ , not the father. So, if $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z)
    otin R$ . R is not transitive.
  Therefore, this relation is neither reflexive, nor symmetric, nor transitive.

Common mistakes

Confusing the conditions for symmetry and transitivity.
Incorrectly assuming a relation is reflexive, symmetric, or transitive without proper verification.
Errors in checking the conditions for all elements in the given set.
Misinterpreting the definition of the relation itself.

Revision tips

Clearly write down the definitions of reflexive, symmetric, and transitive relations before starting.
For each relation, systematically check all three properties one by one.
Pay close attention to the set on which the relation is defined (e.g., N, Z, A).
When disproving a property, provide a specific counterexample.
Practice with diverse examples, including those involving real-world scenarios.

Practice MCQs

Q1. Which property requires that if (a, b) is in the relation R, then (b, a) must also be in R?

Q2. A relation R on a set A is reflexive if:

Q3. If $R = {$ (x, y): y is divisible by x} on the set $A = {1$ , 2, 3, 4}, which property does R satisfy?

Q4. Consider the relation $R = {$ (x, y): x and y work at the same place}. Is this relation transitive?

Q5. The relation $R = {$ (x, y): x - y is an integer} on the set of integers (Z) is:

Frequently asked questions

What are the three main properties of relations discussed in Chapter 1?