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Relations and Functions Class 12th Notes - Free NCERT Class 12 Maths Chapter 1 Notes - Download PDF

Relations and Functions Class 12th Notes - Free NCERT Class 12 Maths Chapter 1 Notes - Download PDF

Edited By Ravindra Pindel | Updated on Apr 23, 2022 01:49 PM IST

Relations and Functions is an important concept in mathematics. It also has good weightage in CBSE Class 12 Board Exam as well as in JEE Main Entrance exam. The students who wish to perform well in the exam need be to thorough with this topic. In this article, you will get NCERT Class 12 Maths chapter 1 notes which will help you to understand the concepts and score well in the exam. Relations and Functions Class 12 notes are designed for the students to get conceptual clarity. Also, these Class 12 Maths chapter 1 notes can be used as revision notes before the exam.

The NCERT Class 12 Maths chapter 1 notes will give a short description of the topics covered in the NCERT Book.

Also, see,

Relation:

Two sets A and B are related if there is a recognizable connection or link between the elements sets A and B. Let ‘a’ is an element of set A and ‘b’ is an element of set B. If (a, b) \in R we can say that a is related to b under the relation R. We can write as aRb to describe the relation.

Let a relation R is defined as R:A\rightarrow B.

Example-. If A = {1, 2, 3} and B = {a, b, c}, then R = {(1, b), (2, c), (1, a), (3, a)} being a subset of A X B, is a relation from A to B. Here (1, b), (2, c), (1, a) and (3, a) belongs to R, so we write 1 Rb, 2Rc, 1Ra and 3Ra.

Total Number of Relations: Let A and B be two non-empty finite sets consisting of m and n elements respectively. Then A x B consists of mn ordered pairs. So, the total number of subsets of A x B is 2mn.

Domain:- The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R.

Domain of R\{\mathrm{a}:(\mathrm{a}, \mathrm{b}) \in \mathrm{R}\}

Range:- The set of all second elements in a relation R from a set A to a set B is called the range of the relation R.

Range of R \{\mathrm{b}:(\mathrm{a}, \mathrm{b}) \in \mathrm{R}\}

Co-domain:- The whole set B is called the co-domain of the relation R.


Note- Range subset of co-domain.

Types of Relation:

Empty Relation:- No element of A is related to any element of A . i.e. \mathrm{R}=\phi \subset \mathrm{A} \times \mathrm{A}.

Universal relation:- Each element of set A is related to every element of set A i.e. \mathrm{R}=\mathrm{A} \times \mathrm{A}

Note- Both empty relation and universal relation are called trivial relation.

Identity Relation: A relation R on A is called the identity relation if every element of A is related to itself only. Let A be a set. Then the relation R = {(a, a) : a belongs to A} on A is called the identity relation on A.

Reflexive Relation:- A relation R on a set A is said to be reflexive if every element of A is related to itself. If (a, a) \in \mathrm{R} for every a \in \mathrm{A}.

Note : Every Identity relation is reflexive but every reflexive ralation is not identity.
Symmetric Relation:- if\left(a_{1}, a_{2}\right) \in \mathrm{R}\Rightarrow \left(a_{2}, a_{1}\right) \in \mathrm{R} for all a_{1}, a_{2} \in \mathrm{A}Transitive Relation:- if \left(a_{1}, a_{2}\right) \in \mathrm{R} and \left(a_{2}, a_{3}\right) \in \mathrm{R} \Rightarrow \left(a_{1}, a_{3}\right) \in \mathrm{R} for all a_{1}, a_{2}, a_{3} \in \mathrm{A}

Equivalence Relation:- If a relation R is reflexive, symmetric, and transitive then it is said to be an equivalence relation.

Function:

The function is a binary relation between two sets A and B that associates each element of set A to exactly one element of set B. So, If a vertical line cuts a given graph at more than one point then it can not be the graph of a function.

Note - Every function is a relation but every relation is not necessarily a function.

Let a function 'f' is defined asf: A \rightarrow B.

Domain:- The set A is called the domain of the function.

Co-domain:- The set B is called the co-domain of the function.

Range:- The set of 'f' images of all the elements of set A is known as the range of the function 'f'.

Types of Function:-

Injective function:- A function f is called injective if the images of distinct elements of A under f are distinct. If function f: \mathrm{A} \rightarrow \mathrm{B} is defined for everya_{1}, a_{2} \in \mathrm{A}, f\left(a_{1}\right)=f\left(a_{2}\right)\Rightarrow a_{1}=a_{2}. then this is one-one function.

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Many-one function:- A function f is called many-one if the images of distinct elements of A under f can be same.

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surjective function:- A function f:X\rightarrow Y is called surjctive if every element of Y is the image of some element of X.

fireshot-capture-196-

Composition of Functions and Invertible Function:-

Composite function:-

If function f: A \rightarrow B and g: B \rightarrow Care two functions then the composite function of two functions 'f' and 'g' is gof : A \rightarrow C is defined as gof(x)=g(f(x)) \forall x \in A .

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Example-

If f (x) = 8x^{3} and g(x) = x^{\frac{1}{3}} than find f(g(x)) and g(f(x))

Solution:

The solution is as follows

(ii) f (x) = 8x^{3} and g(x) = x^{\frac{1}{3}}

gof = g(f(x))

= g( 8x^{3})

= ( 8x^{3})^{\frac{1}{3}}

=2x

fog = f(g(x))

=f(x^{\frac{1}{3}} )

=8((x^{\frac{1}{3}} )^{3})

=8x

Inverse function:-

If function f:\mathrm{A} \rightarrow \mathrm{B} be a one-one and onto function then there exists a unique function \mathrm{g}: \mathrm{B} \rightarrow \mathrm{A} such that f(x)=y. \Leftrightarrow g(y)=x \forall x \in A \ \ \& \ \ y \in B. The function g is the inverse of f.

g=\mathrm{f}^{-1}: \mathrm{B} \rightarrow \mathrm{A}.

Example-

If function f (x) = 4x + 3 is defined f : R \rightarrow R than find the inverse of f.

Solution:

f (x) = 4x + 3

y=4x+3\, \, \, , y \in R

\Rightarrow x=f^{-1}(y) = \frac{y-3}{4} \in R

Algebraic Operations on Functions:-

Let two real-valued fuctions f & g are given with domain set A, B respectively.

(i) (f + g ) ( x) = f(x) + g(x) (domain is \mathrm{A} \cap \mathrm{B} )

(ii) (f - g ) ( x) = f(x) - g(x) (domain is \mathrm{A} \cap \mathrm{B} )

(iii) (f . g ) ( x) = f(x) g(x) (domain is \mathrm{A} \cap \mathrm{B} )

(iv) (f/g ) ( x) = f(x)/g(x) (domain is \mathrm{A} \cap \mathrm{B} - { x | g(x) = 0 } )

Binary Operations:-

Binary operation * on a set A is function * : A \times A \rightarrow A. Binary operation * is denoted * (a,b) as a*b.

Commutative property:- If a*\mathrm{b}=\mathrm{b} * \mathrm{a} for every \mathrm{a} ,\mathrm{b} \in \mathrm{A} than * is commutative on A.

Associative property:- If (a*\mathrm{b})*c=a*(\mathrm{b} * \mathrm{c}) for every \mathrm{a} ,\mathrm{b},c \in \mathrm{A} than * is associative on A.

Identity:- If an element e in set A exists such that a * \mathrm{e}=\mathrm{a}=\mathrm{e} * \mathrm{a}, \forall \mathrm{a} \in \mathrm{A} than 'e' is an identity element for * on set A.

Note- Zero is the identity for addition operation on R but not on N as 0 doesn't belong to N.

NCERT Class 12 Notes Chapter Wise.

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Relations and Functions Class 12 Notes-Important Points to Remember

  • First, try to solve all the NCERT exercises on your own. You can take help from these CBSE Class 12 Maths chapter 1 notes which will help you to understand the concepts easily.

  • If you solved all NCERT problems then you can try to solve the CBSE 12 Board Previous Year's Papers.

  • You can use Class 12 Maths chapter 1 notes pdf download offline as quick revision notes before the exam.

    Happy learning !!!

Frequently Asked Question (FAQs)

1. What is the weightage of the chapter Relations and Functions for CBSE board exam ?

The total weightage of Relations and Functions is 4-5 marks in the final board exam.

2. Which are the most difficult chapters of NCERT Class 12 Maths syllabus?

Some students consider Probability and Integration are the most difficult chapter in the CBSE Class 12 Maths.

3. How does the NCERT Notes are helpful in the board exam ?

NCERT notes are provided in a very simple language, so they can be understood very easily. These notes can be used to revise important concepts.

4. Does CBSE provides the revision notes for NCERT Class 12?

No, CBSE doesn't provide any short notes or revision notes for any Class.

5. What are the various types of relations ?

Empty Relation, Reflexive Relation, Symmetric Relation, Transitive Relation, Anti-symmetric Relation, Universal Relation, Inverse Relation, and Equivalence Relation are some important types of relation.

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