Relations and Functions Class 12 Notes CBSE Maths Chapter 1 (Free PDF Download)

Section–A (1 Mark Questions)

1. Define Empty relation.

Ans. For any set A, an empty relation defined on A as: there is no element exists in the relation set which satisfies the relation for a given set A.

2. If $f(x)=\frac{x-\left | x \right |}{\left | x \right |}$, then find f (-1). 

Ans. $f(x)=\frac{x-\left | x \right |}{\left | x \right |}$ 

$\therefore f(-1)=\frac{-1-\left | 1 \right |}{\left | 1 \right |}=\frac{-1-1}{1}=2$

3. If n(A) = p and n(B) = q, then find the number of relations from set A to set B.

Ans. Number of the relation from set A to set B is given by 2n(A) × n(B)

4. A function is called an onto function, if its range is equal to ________. 

Ans. From the definition of onto. 

5. A binary operation * on a set X is said to be ________, if a * b = b * a, where a, b ∈ X.

Ans. Commutative

Section–B (2 Marks Questions)

6. The domain of the function $f:R\rightarrow R$ defined by $f(x)=\sqrt{x^{2}-3x+2}$  is ________.

Ans. Here $x^{2}-3x+2\geq 0$ 

$\Rightarrow (x-1)(x-2)\geq 0$

$\Rightarrow x\leq or x\geq2$

Hence the domain of $f=(-\infty ,1]\cup [2,\infty ])$ .

7. Find the domain of the real valued function f defined by $f(x)=\sqrt{(25-x^{2})}$ .

Ans. Given, $f(x)=\sqrt{(25-x^{2})}$ 

The function is defined if $25-x^{2}\geq 0$

$\Rightarrow x^{2}\leq 25$

$\Rightarrow -5\leq x\leq 5$

Therefore, the domain of the given function is [-5,5].

8. Consider f: {1, 2, 3} → {a, b, c} given by f(1) = a,  f(2) = b and f(3) = c. Find f-1 and show that (f--1)-1 = f.

Ans. f = {(1, a) (2, b) (3, c)}

f-1 = {(a, 1) (b, 2) (c, 3)}

(f -1) -1 = {(1, a) (2, b) (3, c)}

Hence (f-1)-1 = f.

9. Let R be the relation on N defined as xRy if x + 2y = 8. Find the domain of R.

Ans. As xRy if x + 2 y = 8. 

Therefore, domain of the relation R is given by x = 8 – 2y ∈ N. 

When y = 1 ⇒ x = 6. 

When y = 2 ⇒ x =4.

When y = 3 ⇒ x = 2. 

When y = 4 ⇒ x = 0 ∉ N. 

Therefore, domain is {2, 4, 6}.

10. What is the range of the function $f(x)\frac{\left | x-1 \right |}{x-1},x\neq 1\;?$

Ans. Given, function is $f(x)\frac{\left | x-1 \right |}{x-1},x\neq 1$  The above function may be written as 

$\begin{array}{l}f(x)=\left\{\begin{array}{ll}\frac{x-1}{x-1}, & \text { if } x>1 \\ -\frac{(x-1)}{x-1}, & \text { if } x<1\end{array}\right. \\ \Rightarrow f(x)=\left\{\begin{array}{ll}1, & \text { if } x>1 \\ -1, & \text { if } x<1\end{array}\right.\end{array}$

$\therefore$ Range of f(x) is the set {-1, 1}.

11. If f is an invertible function, defined as $f(x)=\dfrac{3x-4}{5}$ then write $f^{-1}(x)$.

Ans. Given $f(x)=\dfrac{3x-4}{5}$  and is invertible.

$\text { Let } y=\frac{3 x-4}{5} \\$

$\Rightarrow 5 y=3 x-4 \\$

$\Rightarrow 3 x=5 y+4 \\$

$\Rightarrow x=\frac{5 y+4}{3} \\$

$\Rightarrow f^{-1}(y)=\frac{5 y+4}{3} \\$

$\Rightarrow f^{-1}(x)=\frac{5 x+4}{3}$

12. If R = {(a, a3): a is a prime number less than 5} be a relation. Find the range of R.

Ans.  Given, R = {(a, a3): a is a prime number less than 5} 

We know that 2 and 3 are the prime numbers less than 5.

$\therefore$  a can take values 2 and 3.

Then, R = {(2, 23), (3, 33)} = {(2, 8), (3, 27)}

Hence, the range of R is {8, 27}.

13. If A = {1, 2, 3}, B = {4, 5, 6, 7} and f = {(1, 4), (2, 5), (3, 6)} is a function from A to B. State whether f is one-one or not.

Ans. Given, A = {1, 2, 3} and B = {4, 5, 6, 7} 

Now, f : A $\rightarrow$  B is defined as 

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

Therefore, f(1) = 4, f(2) = 5 and f(3) = 6.

It is seen that the images of distinct elements of A under f are distinct. 

So, f is one-one.

PDF Summary - Class 12 Maths Relations and Functions Notes (Chapter 1)

Relation

• It defines the relationship between two sets of values, let say from set A to set B.

• Set A is then called domain and set B is then called codomain.

• If $\left( a,b \right)\in R$, it shows that $a$ is related to $b$ under the relation $R$

Types of Relations

1. Empty Relation: 

• In this there is no relation between any element of a set. 

• It is also known as void relation 

• For example: if set A is $\left\{ 2,4,6 \right\}$ then an empty relation can be $R=\left\{ x,y \right\}$where $x+y>11$ 

2. Universal Relation:

• In this each element of a set is related to every element of that set.

• For example: if set A is $\left\{ 2,4,6 \right\}$ then a universal relation can be $R=\left\{ x,y \right\}$where $x+y>0$

3. Trivial Relation: Empty relation and universal relation is sometimes called trivial relation.

4. Reflexive Relation: 

• In this each element of set (say) A is related to itself i.e., a relation R in set A is called reflexive if \[\left( a,a \right)\in R\] for every $a\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,1 \right),\left( 1,2 \right),\left( 2,2 \right),\left( 2,1 \right),\left( 3,3 \right) \right\}$ is reflexive since each element of set A is related to itself.

5. Symmetric Relation:

• A relation R in set A is called symmetric if \[\left( a,b \right)\in R\]and $\left( b,a \right)\in R$for every $a,b\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,2 \right),\left( 2,1 \right),\left( 2,3 \right),\left( 3,2 \right),\left( 3,1 \right),\left( 1,3 \right) \right\}$ is symmetric.

6. Transitive Relation:

• A relation R in set A is called transitive if \[\left( a,b \right)\in R\]and $\left( b,c \right)\in R$then $\left( a,c \right)$ also belongs to R for every $a,b,c\in A$.

• For example: if $SetA=\left\{ 1,2,3 \right\}$ then relation $R=\left\{ \left( 1,2 \right),\left( 2,3 \right),\left( 1,3 \right)\left( 2,3 \right),\left( 3,2 \right),\left( 2,2 \right) \right\}$ is transitive.

7. Equivalence Relation:

• A relation $R$ on a set A is equivalent if $R$ is reflexive, symmetric and transitive.

• For example:$R=\left\{ \left( {{L}_{1}},{{L}_{2}} \right):line{{L}_{1}}is parallel line{{L}_{2}} \right\}$, 

This relation is reflexive because every line is parallel to itself

Symmetric because if ${{L}_{1}}$ parallel to ${{L}_{2}}$ then ${{L}_{2}}$ is also parallel to ${{L}_{1}}$

Transitive because if ${{L}_{1}}$ parallel to ${{L}_{2}}$ and ${{L}_{2}}$ parallel to ${{L}_{3}}$ then ${{L}_{1}}$ is also parallel to ${{L}_{3}}$

Functions

• A function f from a set A to a set B is a rule which associates each element of set A to a unique element of set B.

• Set A is domain and set B is codomain of the function 

• Range is the set of all possible resulting values given by the function.

• For example: ${{x}^{2}}$ is a function where values of $x$ will be the domain and value given by ${{x}^{2}}$ is the range.

Types of Function:

1. One-One Function: 

• A function f from set A to set B is called one-one function if no two distinct elements of A have the same image in B.

• Mathematically, a function f from set A to set B if $f\left( x \right)=f\left( y \right)$ implies that $x=y$ for all $x,y\in A$.

• One-one function is also called an injective function.

• For example: If a function f from a set of real numbers to a set of real numbers, then $f\left( x \right)=2x$ is one-one function.

2. Onto Function:

• A function f from set A to set B is called onto function if each element of set B has a preimage in set A or range of function f is equal to the codomain i.e., set B.

• Onto function is also called surjective function.

• For example: If a function f from a set of natural numbers to a set of natural numbers, then $f\left( x \right)=x-1$ is onto the function.

3. Bijective Function:

• A function f from set A to set B is called a bijective function if it is both one-one function and onto function.

• For example: If a function f from a set of real numbers to a set of real numbers, then $f\left( x \right)=2x$ is one-one function and onto function.

Binary Operations

• A binary operation are mathematical operations such as addition, subtraction, multiplication and division performed between two operands.

• A binary operation on a set A is defined as operations performed between two elements of set A and the result also belongs to set A. Then set A is called binary composition.

• It is denoted by $*$ 

• For example: Binary addition of real numbers is a binary composition since by adding two real numbers the result will always be a real number.

Properties of Binary Composition:

• A binary operation $*$ on the set X is commutative, i.e., $a*b=b*a$, for every $a,b\in X$ 

• A binary operation $*$ on the set X is associative, i.e., \[a*\left( b*c \right)=\left( a*b \right)*c\], for every $a,b,c\in X$

• There exists identity for the binary operation $*:A\times A\to A$, i.e., $a*e=e*a=a$ for all $a,e\in A$ 

• A binary operation $*:A\times A\to A$ is said to be invertible with respect to the operation $*$ if there exist an element $b$ in $A$ such that $a*b=b*a=e$, $e$ is identity element in $A$ then $b$ is the inverse of $a$ and is denoted by ${{a}^{-1}}$ 

Relations and Functions Class 12 Notes Mathematics

All the topics and subtopics which are covered in Relations and Functions for Class 12 are given below:

• Introduction

• Types of Relations

• Types of Functions

• Composition of functions and invertible functions (Not in the current syllabus)

• Binary operations

Let’s discuss the concepts of relation and function in a full detailed manner here in Notes on Relations and Functions Class 12.

What are Relations?

• Relations in Maths is one of the very important topics for the set theory.

• Relations and functions generally tell us about the different operations performed on the sets.

• Relation in Maths can be put into term as a connection between the elements of two or more sets and the sets must be non-empty.

• A relation namely R is formed by a Cartesian product of subsets.

Different Types of Relations in Mathematics

Let’s discuss the different types of relations included in Notes on Relations and Functions Class 12. There are different types of relations in mathematics that will help define the connection between the sets. There are eight types of relations in mathematics.

Here Are the Types of Relations in Mathematics

What is a Function?

A function can have the same range mapped as that of in relation, such that a set of inputs is related to exactly one output.

For example Set A & Set B are related in a manner that all the elements of Set A are related to exactly one element of Set B or many different elements of the given set A are related to one element of given  Set B. Therefore this type of the relation is also known as a function.

We can see that any given function cannot have one to Many Relation between the set A and set B.

Types of Functions

In terms of relations, we can define the types of functions as the following:

1. One to One Function

• Let there be a function f: A → B is said to be One to One if for each value of A there is a distinct value of B.

• The one-to-one function is also known as the Injective function.

2. Many to One Function

• A many to one function is one that maps two or more elements of A to the same element of set B.

3. Onto Function 

• A function for which every element of set B there is preimage in set A is known as Onto Function

• The onto function is also known as the Subjective function.

4. One-One and Onto Function

• The function f matches with each element of A with a discrete element of B and every element of B has a preimage in A.

• The one-one and onto function is also known as the Bijective function.

Relation as a Function

A special kind of relation (a set of ordered pairs) which follows a rule that every value of let’s suppose set X must be associated with only one value of Y is known as a Function.

Why are Revision notes on Relations and Functions Class 12 Important?

The learning process is student-specific, that is some students prefer kinesthetic learning, some are auditory learners, and others may find visual learning to be more effective. But all these different processes are just a small part of the learning experience; the other important part is the revision by the students. Students have to keep in touch with the subject and retain their learning; So revision is a good way to bring about this phase.

• Enables the student to reinforce their learning

• Exam stress and anxiety are reduced

• Reduced chances of making simple, but conspicuous mistakes

• Saves precious time during examinations

• The accuracy of answers are higher

• Students become more confident during exams