# NCERT Solutions for Class 12 Maths Chapter 1 Free PDF Download

## NCERT Solutions for Class 12 Maths for Chapter 1 – Relations and Functions:

NCERT Solutions for Class 12 Maths Chapter 1 will provide the basis of two important topics of mats i.e. Relations and Functions. Our team of expert teachers of maths has done hard work to prepare these NCERT Solutions.

NCERT Solutions for Class 12 Maths for chapter 1 Relations and Functions are very helpful to build a strong base of students in these two fundamental topics. These solutions will provide comprehensive solutions for each and every problem of this chapter. It also helps the students to complete their homework properly.

NCERT solutions for class 12 Maths for Chapter 1 Relations and Functions is exactly as per the curriculum of CBSE. Toppr will help you to get complete NCERT solutions. We are providing you the free pdf download links of the NCERT solutions for class 12 maths for chapter 1 Relations and Functions.

### CBSE Class 12 Maths Chapter 1 – Relations and Functions NCERT Solutions

The concept of the term â€˜relationâ€™ in maths came from the meaning of relationships in the English language, which says that two objects or quantities are related if there is a recognizable connection exists between them. NCERT solutions for class 12 maths for chapter 1 will help the students to understand this topic to grasp the theory of relations as well as of functions.

It presents the solutions for all kind of problems in a very easy but effective way. In this chapter, the students will learn types of relations and functions, various mathematical operations on these and also some important theorems. Also, this chapter presents the composition of functions, invertible functions and binary operations in depth.

### Sub-topics covered under NCERT Solutions for Class 12 Maths Chapter 1

• 1.1 Introduction
• 1.2 Types of Relations
• 1.3 Types of Functions
• 1.4 Composition of Functions and Invertible Function
• 1.5 Binary Operations

### NCERT Solutions for Class 12 Maths Chapter 1

NCERT solutions for class 12 Maths for Chapter 1 Relations and Functions elaborate each concept sequentially from relations to functions. Various theorems and operations based on them will give the real concept and inner view of the topic. This chapter explains all major aspects with the help of real-life examples and Vein diagram. Plotting of the functions will give the exact presentations about functions between the independent variable and dependent variable.

Let us discuss the sub-topics in detail.

1.1 Introduction

In the previous class, students have got concepts about the notion of relations and functions, domain, co-domain, and range along with different types of specific real-valued functions and their graphs. Now here they will get expanded concepts in that continuation. Also, types of relations and functions will give much strong base.Â

1.2 Types of Relations

This topic gives definitions and examples of various relations such as Empty relation, Universal relation, reflexive relation, symmetric relation, and transitive relations. Also, Equivalence relation will give a combined view of reflexive, symmetric and transitive relations. Different notations will elaborate on the questions and their solutions. The student will learn to prove various theorems and lemmas.

1.3 Types of Functions

In class XI students has studied the notion of a function and some functions like identity function, constant function, polynomial function, rational function, modulus function, etc. They also have studied their graphs.Â

Now, this chapter will discuss Addition, subtraction, multiplication, and division of two functions. Also, it explores different types of functions like into, onto, one-one, etc. The student will prove the injectivity and surjectivity also.

1.4 Composition of Functions and Invertible Function

In this section, the student will study the composition of functions and the inverse of a bijective function.Â This topic will explain another concept i.e. invertible function.

1.5 Binary Operations

It explains the binary operations which need two operands for it. Any binary operation âˆ— on a set A is given as a function âˆ— : A Ã— A â†’ A. Here we denote âˆ— (a, b)Â  by a âˆ— b.Â A binary operation follows the rule of symmetry, commutativity, and associativity for addition and multiplication both.

### Solved Questions for You

Question 1: Show that the relativeÂ RÂ in the setÂ {1,2,3}Â given byÂ R={(1,2),(2,1)}Â is symmetric but neither reflexive nor transitive.

A relationÂ RÂ onÂ AÂ is defined asÂ R={(1,2),(2,1)}.
It is seen thatÂ (1,1),(2,2),(3,3)âˆˆ/â€‹R.
âˆ´RÂ is not reflexive.
Now, asÂ (1,2)âˆˆRÂ andÂ (2,1)âˆˆR, thenÂ RÂ is symmetric.
Now,Â (1,2)Â andÂ (2,1)âˆˆR
However,
(1,1)âˆˆ/â€‹R
âˆ´RÂ is not transitive.
Hence,Â RÂ is symmetric but neither reflexive nor transitive.

Question 2: Given an example of a relation. Which isÂ Reflexive and symmetric but not transitive.

Define a relationÂ RÂ onÂ AÂ as:
A={(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)}
RelationÂ RÂ is reflexive since for everyÂ {aâˆˆA,(a,a)âˆˆRi.e.,(4,4),(6,6),(8,8)}âˆˆR

RelationÂ RÂ is symmetric sinceÂ (a,b)âˆˆRâ‡’(b,a)âˆˆRÂ for all

a,bâˆˆR.

RelationÂ RÂ is not transitive sinceÂ (4,6),(6,8)âˆˆR, butÂ (4,8)âˆˆ/â€‹R.

Hence, relationÂ RÂ is reflexive and symmetric but not transitive.

Question 3: RelationÂ RÂ in the setÂ AÂ of human beings in a town at aÂ particular time given byÂ R={(x,y):xisfatherofy}

enter 1-reflexive and transitive but not symmetric

Â  Â  Â  Â  Â 2-reflexive only

Â  Â  Â  Â  Â 3-Transitive only

Â  Â  Â  Â  Â 4-Equivalence

Â  Â  Â  Â  Â 5-Neither reflexive, nor symmetric, nor transitive

Answer: R={(x,y):xÂ isÂ theÂ fatherÂ ofÂ y}
(x,x)âˆˆ/â€‹R
AsÂ xÂ cannot be the father of himself.
âˆ´RÂ is not reflexive.
Now, letÂ (x,y)âˆˆR
â‡’xÂ is the father ofÂ y.
â‡’yÂ cannot be the father ofÂ y.
Indeed,Â yÂ is the son or the daughter ofÂ y.
âˆ´(y,x)âˆˆ/â€‹R
âˆ´RÂ is not symmetric.
Now, letÂ (x,y)âˆˆRÂ andÂ (y,z)âˆˆR.
â‡’xÂ is the father ofÂ yÂ andÂ yÂ is the father ofÂ z.
â‡’xÂ is not the father ofÂ z.
Indeed, xÂ is the grandfather ofÂ z.
âˆ´(x,z)âˆˆ/â€‹R
âˆ´RÂ is not transitive.
Hence,Â RÂ is neither reflexive, nor symmetric, nor transitive.

Question 4: LetÂ A={1,2,3},Â B={4,5,6,7} and letÂ f=(1,4),(2,5),(3,6)Â be a function fromÂ AÂ toÂ B. Show thatÂ fÂ is one-one.

Answer: It is given thatÂ A={1,2,3},Â B={4,5,6,7}.

f:Aâ†’BÂ is defined asÂ f=(1,4),(2,5),(3,6).
âˆ´f(1)=4,f(2)=5,f(3)=6
It is seen that the images of distinct elements ofÂ A

underÂ fÂ are distinct

Hence, functionÂ fÂ is one-one.

### Download Toppr – Best Learning App for Class 5 to 12

NCERT Solutions for Class 12 Maths for Chapter 1 Relations and Functions is giving the solutions of the first chapter of maths in a very interesting way. This solution with easy presentations will help students to understand it and also apply to real-life applications.Â

Our App also provides free pdf downloads, free online classes, free video lectures, and free doubt clearing sessions. Our Maths faculties are highly capable and proficient. Download Toppr app forÂ AndroidÂ andÂ iOSÂ orÂ SignupÂ for free.

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