NCERT Solutions for Class 12 Maths Chapter 1

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.

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.

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. 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 ∗ on a set A is given as a function ∗ : A × A → A. Here we denote ∗ (a, b)  by a ∗ b. 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 relative R in the set {1,2,3} given by R={(1,2),(2,1)} is symmetric but neither reflexive nor transitive.

Answer: Let A={1,2,3}
A relation R on A is defined as R={(1,2),(2,1)}.
It is seen that (1,1),(2,2),(3,3)∈/​R.
R is not reflexive.
Now, as (1,2)∈R and (2,1)∈R, then R is symmetric.
Now, (1,2) and (2,1)∈R
However,
(1,1)∈/​R
R is not transitive.
Hence, R is symmetric but neither reflexive nor transitive.

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

Answer: Let A={4,6,8}

Define a relation R on A as:
A={(4,4),(6,6),(8,8),(4,6),(6,4),(6,8),(8,6)}
Relation R is reflexive since for every {aA,(a,a)∈R i.e.,(4,4),(6,6),(8,8)}∈R

Relation R is symmetric since (a,b)∈R⇒(b,a)∈R for all a,bR.

Relation R is not transitive since (4,6),(6,8)∈ R, but (4,8)∈/​R.

Hence, relation R is reflexive and symmetric but not transitive.

Question 3: Relation R in the set A of human beings in a town at a particular time given by R={(x,y):x is father of y}

 1-reflexive and transitive but not symmetric

         2-reflexive only

         3-Transitive only

         4-Equivalence

         5-Neither reflexive, nor symmetric, nor transitive

Answer: R={(x,y):x is the father of y}
(x,x)∈/​R
As x cannot be the father of himself.
R is not reflexive.
Now, let (x,y)∈R
x is the father of y.
y cannot be the father of y.
Indeed, y is the son or the daughter of y.
∴(y,x)∈/​R
R is not symmetric.
Now, let (x,y)∈R and (y,z)∈R.
x is the father of y and y is the father of z.
x is not the father of z.
Indeed, x is the grandfather of z.
∴(x,z)∈/​R
R is not transitive.
Hence, R is neither reflexive, nor symmetric, nor transitive.

Question 4: Let A={1,2,3}, B={4,5,6,7} and let f=(1,4),(2,5),(3,6) be a function from A to B. Show that f is one-one.

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

f:AB is defined as f=(1,4),(2,5),(3,6).
f(1)=4,f(2)=5,f(3)=6
It is seen that the images of distinct elements of A

under f are distinct

Hence, function f is one-one.

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