Relations and Functions

In our day to day life very often we come across questions like, "How is he related to you?". Some probable answers maybe "He is my father", "He is my brother", etc. From this, we see that the word relation connects a person with another person. Extending this in mathematics, we consider relations as one which connects mathematical objects. Function is a concept derived from relations. This article is about the concept of relation and function class 11.

What are Relations and Functions?

A relation in mathematics is a connection or association between elements of two sets. This is crucial for checking the common terms or anything between two or more functions. The number of possible relations between two sets $A$ and $B$ depends on the sizes of these sets. Relation is one of the fundamental concepts for Functions. The main applications of relations and functions are in database systems, graph theory, and social networks.

Relation Definition

Relation is defined as the relation between two different sets of information. Suppose we are given two sets containing two different values then a relation defined such that it connects the value of the first set with the value of the second set is called the relation.

A relation $R$ from a non-empty set $A$ to a non-empty set $B$ is a subset of the cartesian product $A × B$.

The subset is derived by describing the relationship between the first element and the second element of the ordered pairs in $A × B$.

The second element is called the image of the first element.

If the element $(a,b)$ belongs to $R$, (here $a$ belongs to $A$ and $b$ belongs to $B$), then the relation is represented as $a R b$.

Example: Let $A=\{1,2\}$ and $B=\{x,y\}$. A possible relation $R$ could be: $R=\{(1,x),(2,y)\}$. This indicates that $1$ is related to $x$ and $2$ is related to $y$.

Domain of a Relation

The domain of a relation is the set of all first elements (or components) of the ordered pairs in the relation. In other words, it is the set of all possible inputs with at least one associated output in the relation.

The domain of a relation R is the set of all first elements of the ordered pairs in a relation R.

Range of a Relation

The range of any Relation is the set of output values of the relation. For example, if we take two sets A and B, and define a relation R: {(a,b): a ∈ A, b ∈ B} then the set of values of B is called the domain of the function.

The range of a relation R is the set of all second elements of the ordered pairs in a relation R.

eg. $R=\{(1,c),(5,a),(8,b)\}$, then the domain is $\{1,5,8\}$ and the range is $\{c,a,b\}$

Function Definition

A relation from a set $A$ to a set $B$ is said to be a function from $A$ to $B$ if every element of set $A$ has one and only one image in set $B$.

OR

$A$ and $B$ are two non-empty sets, then a relation from $A$ to $B$ is said to be a function if each element $x$ in $A$ is assigned a unique element $f(x)$ in $B$ and no two elements in set $B$ are mapped to the same element in set $A$ , and it is written as

$f: A ➝ B$ and read as $f$ is mapping from $A$ to $B$.

First and second images represent a function as all the elements in set $A$ is mapped a element in set $B$ while the element $d$ in set $A$ is not mapped to any element in set $B$.

A function can be said as a subset of relation. Every functions are relations but not every relations are functions.

Image of a function

The image of a function refers to the set of all output values it produces from its domain.

Given a function $\mid f: A \rightarrow B$ and a subset $X \subseteq A$, the image of $X$ under $f$ is the set of all elements $f(x)$ where $x \in X$. The image of $X$ is denoted as $f(X)$ and is defined as:

$
f(X)=\{f(x) \mid x \in X\}
$

If we consider the entire domain $A$, the image of the function $f$, also called the range, is: Image $(\mathrm{f})=\mathrm{f}(\mathrm{A})=\{(\mathrm{a}) \mid \mathrm{a} \in \mathrm{A}\}$

Pre-image of a function

The pre-image of a function refers to the set of all input values that produce a given output value or set of output values.

Given a function $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and a subset $\mathrm{Y} \subseteq \mathrm{B}$, the pre-image of $Y$ under $f$ is the set of all elements $\mathrm{x} \in \mathrm{A}$ such that $\mathrm{f}(\mathrm{x}) \in \mathrm{Y}$. The pre-image of $Y$ is denoted as $f^{-1}(Y)$ and is defined as: $f^{-1}(Y)=\{x \in A \mid f(x) \in Y\}$ If $y \in B$, the preimage of $\{y\}_{\text {is: }}$

$
f^{-1}(\{y\})=\{x \in A \mid f(x)=y\}
$

If $f$ is a function from $A$ to $B$ and $(\mathrm{a}, \mathrm{b})$ belongs to $f$, then $\mathrm{f}(\mathrm{a})=\mathrm{b}$, where ' $b$' is called the image of ' $a$ ' under $f$ and ' $a$ ' is called the pre-image of ' $b$ ' under $f$.

In the ordered pair $(1,2)$. $1$ is the pre-image of $2$ .

Domain of a Function

The collection of all potential input values for which a function can be defined is known as its domain or All possible values of $x$ for $f(x)$ is defined $(f(x)$ is a real number) is known as a domain.

If a function is defined from $A$ to $B$ i.e. $f: A \rightarrow B$, then all the elements of set $A$ is called the Domain of the function.

Co-domain of a Function

If a function is defined from $A$ to $B$ i.e. $f: A \rightarrow B$, then set $B$ is called the Codomain of the function.

Codomain is the set of the values including the range of the function nd it can have some additional values. The range is the Subset of the Codomain.

Range of a Function

The set of all possible values of $f(x)$ for every $x$ belonging to the domain is known as the Range of this function.

The range of a function is the set of all the outputs of the function. For any function $f: A \rightarrow B$ the sets of values in the $B$ are the range of the function. If $f: A \rightarrow B$ is a function such that $f(x)=x^2$ and $A$ is the set of all integers then the range of the function is the set of Range $=\{1,4,9,16, \ldots\}$. We have to note that the range of the function is the subset of the Co-Domain of the function.

For example, let $A=\{1,2,3,4,5\}$ and $B=\{1,4,8,16,27,64,125\}$. The function $f: A->B$ is defined by $f(x)=x^3$. So here,

Domain: Set $A$
Co-Domain: Set $B$
Range: $\{1,8,27,64,125\}$

The range is always a subset of the co-domain and the Range can be equal to the co-domain in some cases.

Representation of Relation and Function

Relations and Functions can be represented in the following ways,

Roster Form

The relations and functions are represented in Roster form as ordered pairs listed explicitly within curly brackets.

Example: Let $A=\{1,2\}$ and $B=\{x,y\}$.
$R=\{(1,x),(2,y)\}$.

Set Builder Form

In set builder form, the relations and functions are represented as a common definition that is not possessed by any element outside the relation or function. The set builder notation are ': or '|' is read as 'such that' respectively.

Example: Let $A = \{1,2,3,4,5\}$ and $B=\{1,2,3,4,5,6,7,8,9\}$
Let $f: A \rightarrow B$ defined as $f(x) = x+1$
$R = \{(x,y): x \in A, y \in B$ and $y = x+1\}$

Arrow Diagram

Using arrow aiagram, the relations and functions can be mapped from one element to the other with the help of arrows.

Example: let $A=\{1,2,3,4,5\}$ and $B=\{1,4,8,16,27,64,125\}$. The function $f: A->B$ is defined by $f(x)=x^3$.

Tabular Form

In this form of representation, the input and output elements are represented in a tabular column.

Example: let $A=\{1,2,3,4,5\}$ and $B=\{1,4,8,16,27,64,125\}$. The function $f: A->B$ is defined by $f(x)=x^3$.

Difference Between Relations and Function

Types of Relations and Functions

Types of Relation

The types of relations are

1. Empty Relation

A relation $R$ on a set $A$ is called an empty relation, if no element of $A$ is related to any element of $A$, i.e., $R = φ$

For example, Let $A = \{2, 4, 6\}$ and $R = \{(a, b) : a, b ∈ A$ and $a + b$ is odd$\} $

Here, $R$ contains no element, therefore $R$ is an empty relation on $A $

2. Universal Relation

A relation $R$ on a set $A$ is called a universal relation, if each element of $A$ is related to every element of $ A$, i.e., $ R$ has all the ordered pair contained in $A \mathrm{x} A$

So, $R = A \mathrm{x} A$.

For example,

1. Let $A = \{2,4\}$ and $R = \{(2,2), (2,4), (4,2), (4,4)\}$

Here, $R = A \mathrm{x} A$. Hence, $R$ is a Universal relation

2. Let $A = \{1,2,3\}$, and $R = \{(a, b) : |a - b| > -2, a , b ∈ A\}$

Clearly the mod value of the difference of any pair $(a,b)$ will be greater than $-2$

So, each possible ordered pair in $A \mathrm{x} A$ will lie in $R$, therefore $R$ is a universal relation

3. Identity relation

If every element of $A$ is related to itself only, then it is known as an identity relation on $A$. It is denoted by $I_A$
$R=\{(a, b): a \in A, b \in A$ and $a=b\}$
It can also be written as $\mathrm{I}_{\mathrm{A}}=\{(\mathrm{a}, \mathrm{a}): \mathrm{a} \in \mathrm{A}\}$
For example $A=\{2,4,6\}$
Then, $I_A=\{(2,2),(4,4),(6,6)\}$

4. Reflexive Relation

A relation $R$ on a set $A$ is called Reflexive, if $(a, a) \in R$, for every $a \in A$,
For example: let $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,1),(2,2),(3,3)\}$
- $\mathrm{R}_2=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3)\}$
- $\mathrm{R}_3=\{(1,1),(2,2),(2,3),(3,2)\}$

Here $R_1$ and $R_2$ are reflexive relations on $A, R_3$ is not a reflexive relation on $A$ as $(3,3)$ is not present in $\mathrm{R}_3$.

5. Symmetric Relation

A relation $R$ on a set $A$ is said to be a symmetric relation, if $a R b \Rightarrow b R a, \forall a, b \in A$
For example, $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,2),(2,1)\}$
- $R_2=\{(1,2),(2,1),(1,3),(3,1)\}$ and
- $\mathrm{R}_3=\{(2,3),(1,3),(3,1)\}$

Here $R_1$ and $R_2$ are symmetric relations on $A$ but $R_3$ is not a symmetric relation on $A$ because $(2,3)$ is in $R_3$ and $(3,2)$ is not in $R_3$.

6. Transitive Relation

A relation $R$ on a set $A$ is said to be a transitive relation, if $a R b$ and $b R c \Rightarrow a R c, \forall a, b, c \in A$
For example, Let $A=\{1,2,3\}$
- $\mathrm{R}_1=\{(1,2),(2,3),(1,3),(3,2)\}$
- $\mathrm{R}_2=\{(2,3),(3,1)\}$
- $\mathrm{R}_3=\{(1,3),(3,2),(1,2)\}$

Here $R_1$ is not a transitive relation on $A$ because $(2,3)$ is in $R_1$ and $(3,2)$ is in $R_1$ but $(2,2)$ is not in $R_1$. Also, $(3,2)$ in $R_1$ and $(2,3)$ is in $R_1$, but $(3,3)$ is not in $R_1$

Again $R_2$ is not a transitive relation on $A$ because $(2,3)$ is in $R_2$ and $(3,1)$ is in $R_2$ but $(2,1)$ is not in $R_2$.
Finally, $R_3$ is a transitive relation.

7. Equivalence Relations

$A$ relation $R$ on a set $A$ is said to be an equivalence relation if $R$ is reflexive, symmetric, and transitive.

Let $R = \{(1,1), (1,2), (2,1), (2,2)\}$

Here, $R$ is a reflexive relation, symmetric relaion and transitive relation. Therefore, $R$ is a equivalence relation.

8. Inverse Relations

Inverse relation occurs when a set has inverse pairs of another set. i.e. if $R \in A \times B$ then the inverse relation is $R^{-1}=\{(b, a)$ such that $(a, b) \in R\}$

Consider if set $A=\{(1,2),(3,4)\}$, then inverse relation will be $R^{-1}=\{(2,1),(4,3)\}$.

Types of Functions

The types of functions include,

1. One-to-One Function(injective)

An injective function, sometimes referred to as a one-to-one function, is one in which distinct elements of $A$ have distinct relationships with $B$ or distinct images with $B$.

A function $f: X \rightarrow Y$ is called a one-one (or injective) function, if different elements of $X$ have different images in $B$. i.e. no two elements of set $X$ can have the same image.
Consider,
$f: X \longrightarrow Y$, function given by $y=f(x)=x$, and
$X=\{-2,2,4,6\}$ and $Y=\{-2,2,4,6\}$

2. Many-to-One Function:

Many one function is a function in which two or more elements of a set are connected to a single element of another set. A function $f: X \rightarrow Y$ is called a many-one function if two or more elements of set $X$ have the same image in set $Y$,

Or we can say that if $f: X \rightarrow Y$ is many- one if it is not one-one function.

Both are many one, as in both there are two elements $x_2, x_3$ which corresponds to the same image $y_3$, i.e. $f\left(x_2\right)=f\left(x_3\right)=y_3$

Eg. Function: $f(x)=x^2$
Domain: All real numbers $(R)$
Codomain: Non-negative real numbers $([0, \infty)$ )

3. Onto Function(Surjective):

A function $f: X \rightarrow Y$ 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 $y \in Y$, there exists an element $x$ in $X$ such that $f(x)=y$

Hence, Range = co-domain for an onto function
Example: Consider, $\mathrm{X}=\left\{\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3, \mathrm{x}_4\right\}$ and $\mathrm{Y}=\left\{\mathrm{y}_1, \mathrm{y}_2, \mathrm{y}_3\right\}$

$
f: X \rightarrow Y
$

4. Bijective Function:

A function $f: X \rightarrow Y$ is said to be bijective, if $f$ is both one-one and onto (both injective and surjective)

Eg.
Consider, $\mathrm{X}_1=\{1,2,3\}$ and $\mathrm{X}_2=\{\mathrm{x}, \mathrm{y}, \mathrm{z}\}$

$
f: X_1 \rightarrow X_2
$

5. Odd and Even Function:

If for a function $f(x), f(-x)=-f(x)$ then the function is known as an odd function. Odd functions are symmetric about the origin.

If for a function $f(x), f(-x)=f(x)$ then the function is known as an even function. Even functions are symmetric about the $y$-axis.

6. Identity function:

The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $y=f(x)=x$ for each $x \in \mathbf{R}$ is called the identity function.
Domain of $f=\mathbf{R}$
Range of $f=\mathbf{R}$

7. Constant function:

The function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $y=f(x)=\mathrm{C}, x \in \mathbf{R}$, where $C$ is a constant $\in \mathbf{R}$, is a constant function.
Domain of $f=\mathbf{R}$
Range of $f=\{\mathrm{C}\}$

8. Polynomial function:

A real valued function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $y=f(x)=a_0$ $+a_1 x+\ldots+a_n x^n$, where $n \in \mathbf{N}$, and $a_0, a_1, a_2 \ldots a_n \in \mathbf{R}$, for each $x \in \mathbf{R}$, is called Polynomial functions.

9. Composite Functions:

Let $\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}$ and $\mathrm{g}: \mathrm{B} \rightarrow \mathrm{C}$ be two functions. Then the composition of $f$ and $g$ is denoted by $g \circ f(x)$ and defined as the function $g \circ f(x)$ : $A \rightarrow C$ given by $g \circ f(x)=g(f(x))$

10. Rational function:

These are the real functions of the type $\frac{f(x)}{g(x)}$, where $f(x)$ and $g(x)$ are polynomial functions of $x$ defined in a domain, where $g(x) \neq 0$. For example $f: \mathbf{R}-\{-2\} \rightarrow \mathbf{R}$ defined by $f(x)=\frac{x+1}{x+2}, \forall x \in \mathbf{R}-\{-2\}$ is a rational function.

11. The Modulus function:

The real function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=|x|=$

$
\begin{aligned}
& x, x \geq 0 \\
& -x, x<0
\end{aligned}
$

$\forall x \in \mathbf{R}$ is called the modulus function.
Domain of $f=\mathbf{R}$
Range of $f=\mathbf{R}^{+} \cup\{0\}$

12. Signum function:

The real function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by

$
f(x)=\left\{\begin{array}{c}
\frac{|x|}{x}, x \neq 0 \\
0, x=0
\end{array}=\left\{\begin{array}{lll}
1, & \text { if } & x>0 \\
0, & \text { if } & x=0 \\
-1, & \text { if } & x<0
\end{array}\right.\right.
$

is called the signum function.
Domain of $f=\mathbf{R}$, Range of $f=\{1,0,-1\}$

13. Greatest integer function:

The real function $f: \mathbf{R} \rightarrow \mathbf{R}$ defined by $f(x)=[x], x \in \mathbf{R}$ assumes the value of the greatest integer less than or equal to $x$, is called the greatest integer function.
Thus

$
\begin{aligned}
f(x)= & {[x]=-1 \text { for }-1 \leq x<0 } \\
f(x)= & {[x]=0 \text { for } 0 \leq x<1 } \\
& {[x]=1 \text { for } 1 \leq x<2 } \\
& {[x]=2 \text { for } 2 \leq x<3 \text { and so on } }
\end{aligned}
$
14. Periodic Function:

A function $f(x)$ is called a periodic function, if there exists a +ve real number $T$ such that $f(x+T)=f(x) \forall x$ belongs Domain of $f(x)$.

Here, $T$ is called the period of $f(x)$, where $T$ is the least $+v e$ value.

Relations and Functions Formulas

Algebra of Functions

Let $f$ and $g$ be two functions. Then,

• $(f+g)(x)=f(x)+g(x) $
• $(f-g)(x)=f(x)-g(x)$
• $(f . g)(x)=f(x) \cdot g(x)$
• $(k f)(x)=k(f(x))$ where $k$ is a real number
• $\{f / g\}(x)=f(x) / g(x), g(x) \neq 0$

List of Topics According to NCERT/JEE MAIN

Importance of Relations and Functions

We come across many relations in our daily lives, such as number x is more significant than number y, triangle m is similar to triangle n, and set A is a subset of set B. See, there is some relationship between the pairs of objects in a specific order in all of these. Thus, Relations and functions are crucial for arithmetic. Practising Class 11 Maths NCERT Chapter 2 will ensure a thorough understanding of essential topics centred on Relations and Functions.

Although, in the JEE test, there are two or three questions from Relation and Functions.
However, it is still very important.
Relation and Function are quite important from the exam point of view. In mathematics, "sets, relations, and functions" is one of the most important topics of set theory. Sets, relations, and functions are three different words having different meanings mathematically but equally important for the preparation of JEE mains.

How to Study Relations and Functions?

Start preparing by understanding and practicing what is relations and functions. Try to be clear on every types of relations and functions along with domain, co-domain and range. Practice drawing graphs for each functions to have better understanding. For every problems related to functions solve it by drawing a mapping graph which could help avoid mistakes.

If you are preparing for competitive exams then solve as many problems as you can. Do not jump on the solution right away. Remember if your basics are clear you should be able to solve any question on this topic.

Important Books for Relations and Functions

Start from NCERT Books, the illustration is simple and lucid. You should be able to understand most of the things. Solve all problems (including miscellaneous problem) of NCERT. If you do this, your basic level of preparation will be completed.

Then you can refer to the book Algebra Arihant by Dr. SK goyal or RD Sharma or Cengage Mathematics Algebra but make sure you follow any one of these not all. Relations and Functions are explained very well in these books and there are an ample amount of questions with crystal clear concepts. Choice of reference book depends on person to person, find the book that best suits you the best, depending on how well you are clear with the concepts and the difficulty of the questions you require.