CBSE Class 12 Maths Notes Chapter 1 Relations and Functions

CBSE Class 12 Maths Notes Chapter 1: Relations and Functions are important for students preparing for their board exams. These notes provide a detailed overview of key concepts, including the definitions, properties, and types of relations and functions.

CBSE Class 12 Maths Notes Chapter 1 Relations and Functions Overview

CBSE Class 12 Maths Notes Chapter 1 PDF

CBSE Class 12 Maths Notes Chapter 1 Relations and Functions PDF

CBSE Class 12 Maths Notes Chapter 1 Relations and Functions

Relations and Functions For Class 12 Concepts

• Introduction
• Types of Relations
• Types of Functions
• Composition of functions and invertible functions
• Binary operations

Relation

CBSE Class 12 Sample Paper

Types of Relations

Empty Relation

Universal Relation

Reflexive- if (a, a) ∈ R , for every a ∈ A,

Symmetric- if (a ₁ , a ₂ ) ∈ R implies that (a ₂ , a ₁ ) ∈ R , for all a ₁ , a ₂ ∈ A,

Transitive- if (a ₁ , a ₂ ) ∈ R and (a ₂ , a ₃ ) ∈ R implies that (a ₁ , a ₃ ) ∈ R for all a ₁ , a ₂ , a ₃ ∈ A.

Equivalence Relation- A relation R in a set A is an equivalence relation if R is reflexive, symmetric and transitive.

Functions

Types of Functions

1. One to one Function: A function f : X → Y is defined to be one-one (or injective), if the images of distinct elements of X under f are distinct, i.e., for every x ₁ , x ₂ ∈ X, f(x ₁ ) = f(x ₂ ) implies x ₁ = x ₂ . Otherwise, f is called many-one.

One-to-One Function Example: Each student in a class having a unique student ID.

Onto Function: A function f: X → Y is said to be onto (or surjective), if every element of Y is the image of some element of X under f, i.e., for every y ∈ Y, there exists an element x in X such that f(x) = y.

One-one and Onto Function: A function f: X → Y is said to be one-one and onto (or bijective), if f is both one-one and onto.

Onto Function Example: A function that assigns students to different classes, ensuring every class has at least one student.

Composition of Functions

Let f: A → B and g: B → C be two functions. Then the composition of f and g, denoted by gof , is defined as the function gof: A → C given by;

gof (x) = g(f (x)), ∀ x ∈ A

Invertible Functions

A function f : X → Y is defined to be invertible if there exists a function g : Y → X such that gof = I _X and fog = I _Y . The function g is called the inverse of f and is denoted by f ^–1 . An important note is that, if f is invertible, then f must be one-one and onto and conversely, if f is one-one and onto, then f must be invertible.

Binary Operations

In mathematics, a binary operation on a set A is a function that combines two elements from the set to produce another element in the same set. Formally, a binary operation ∗ on $A$ is a function: $\ast :A\times A\to A$ This can be denoted as a∗b , where a and b are elements of A , and the result a∗b is also an element of A . Binary operations are fundamental in various fields of mathematics and are used to define algebraic structures such as groups, rings, and fields.

Practice Questions of CBSE Class 12 Maths Notes Chapter 1 Relations and Functions

Here are the Practice Questions of CBSE Class 12 Maths Notes Chapter 1 Relations and Functions-

Question 1: Find out whether each of the following relations are reflexive, symmetric and transitive.

(i) Relation R in the set A = {1, 2, 3…13, 14} defined as

R = {(x, y): 3x − y = 0}

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

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

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

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

Answer:

(i) A = {1, 2, 3 … 13, 14}

R = {(x, y): 3x − y = 0} ∴R = {(1, 3), (2, 6), (3, 9), (4, 12)} R is not reflexive since (1, 1), (2, 2) … (14, 14) ∉ R. Also, R is not symmetric as (1, 3) ∈R, but (3, 1) ∉ R. [3(3) − 1 ≠ 0] Also, R is not transitive as (1, 3), (3, 9) ∈R, but (1, 9) ∉ R. [3(1) − 9 ≠ 0] Hence, R is neither reflexive, nor symmetric, nor transitive.

(ii) R = {(x, y): y = x + 5 and x < 4} = {(1, 6), (2, 7), (3, 8)}

It is seen that (1, 1) ∉ R. ∴R is not reflexive. (1, 6) ∈R But,(6, 1) ∉ R. ∴R is not symmetric. Now, since there is no pair in R such that (x, y) and (y, z) ∈R, then (x, z) cannot belong to R. ∴ R is not transitive. Hence, R is neither reflexive, nor symmetric, nor transitive.

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

R = {(x, y): y is divisible by x} We know that any number (x) is divisible by itself. (x, x) ∈R ∴R is reflexive. Now, (2, 4) ∈R [as 4 is divisible by 2] But, (4, 2) ∉ R. [as 2 is not divisible by 4] ∴R is not symmetric. Let (x, y), (y, z) ∈ R. Then, y is divisible by x and z is divisible by y. ∴z is divisible by x. ⇒ (x, z) ∈R ∴R is transitive. Hence, R is reflexive and transitive but not symmetric.

Benefits of CBSE Class 12 Maths Notes Chapter 1 Relations and Functions

• Clear Understanding of Concepts : The notes provide a structured and detailed explanation of relations and functions making complex mathematical concepts more accessible and easier to understand. This clarity helps students grasp fundamental principles important for advanced topics.
• Comprehensive Coverage : Covering topics such as types of relations, types of functions, composition of functions, and invertible functions, these notes ensure that students receive a thorough overview of Chapter 1. This detailed approach helps in building a strong foundation in mathematics.
• Expert Preparation : Prepared by subject experts, the notes reflect a deep understanding of the curriculum and exam requirements. This expert preparation ensures that the content is accurate, relevant and based on the latest CBSE syllabus.