Relations and Functions : Concepts & Examples

In this post we will specifically discuss relations – functions and its problems.
Contrary to the popular opinion, the concept of relations is easy to understand and practice. Most teacher make this concept complex and difficult, In this post we have made an effort to make the concept of relations easy for you.

Concept of Relations

Relation in Practical life

The meaning of word “Relation” is “something with which you can associate“.

In practical life we come across different relations like
a. Blood Relation: Father-Son relation, Brother – Sister Relation
b. Formal Relationships : teacher – student relation, Guru – devotee relation
c. Characteristics : Different balls of same color

In all the above example, there is some form of relationships between two entity.

Relations and Functions Class 11

Similarly in mathematics, we come across different types of relationships:
a. Number 2 is greater than number 1
b. Lines which are parallel to each other
c. Triangles which are similar to each other

Below you can see relation of color ( Red & Blue ) with balls (P, Q, R, S)

jee main mathematics sets, relations and functions

Each entity above is showing some kind of relationship with other entity.
The study of this relationship is the focus of this chapter. In this post we will try to study the concepts which are to the level of Grade 11.

What are Relations in mathematics

Relation is basically a collection of ordered pair of entity/numbers.
Ordered pair looks like this —> (2, 3}
which means number 2 is related to 3 in some form.

Now the collection of ordered pair would be like:
P = { (1,4), (2, 6), (3, 9), (4, 12) }

The set P consist of 4 ordered pair of numbers and hence is a relation.
This relation is expressed in the form of sets but you can also represent in the form of table, graph or map

Tabular Representation
Here i have expressed the relation P in tabular form

Relations and functions class 11

Coordinate Representation
You can also expressed relation in the graph chart with x and y coordinate

Sets relations and functions

Map representation
This is the most widely used method to show any relation

Relations and functions class 11

Important Parameters of Relations

1. Domain of Relation

All the first elements of ordered pair is known as domain of relation

Let us consider a relation
P = {(2, 5), (1, 3), (4,5), (4, 7), (3,1)}
The first element of the ordered pair are 2, 1, 4, 4, 3 are known as domain of relation

what is relation in math

2. Range of Relation

All the second element of the ordered pair is called Range

Let us again consider the above example
P = {(2, 5), (1, 3), (4,5), (4, 7), (3,1)}

Here P is the relation.
All the second element of the ordered pair are 5, 3, 5, 7, 1. All these numbers are called Range of the relation


Understand that the questions here need the knowledge of set theory. If you are unaware of that chapter, i would strongly recommend to learn that chapter first.

(01) Let A = {1, 2, 3, 4, 5, 6}.
Define a relation R from A to A by R = {(x, y) : y = x + 1 }

(i) Depict this relation using an arrow diagram.
(ii) Write down the domain and range of R.

Here Set A is given as:
A = {1, 2, 3, 4, 5, 6}

Relation is defined as (using set A):
R = {(x, y) : y = x + 1 }

Putting the values of set A in x to get the relation
R = {(1, 2), (2, 3) , (3, 4), (4, 5), (5, 6), (6, 7)}

(i) Depicting this relation in arrow diagram

concepts of relations and functions

(02) A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form

In the questions two sets are given
A = { 1, 2, 3, 5}
B = {4, 6, 9}

we have to define relation such that
R = {(x, y): such that x – y is odd}

The number of such ordered pairs are
R = {(1,4), (1,6), (2,9), (3,4), (3, 6), (5,4), (5,6)}

(03) Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}

The relation for the above condition is
R = {(0, 5), (1,6), (2, 7), (3, 8), (4,9), (5, 10)}

Let us draw the relation in map form

Definition of relations and functions

What is Function in math

The concept of function is very similar to Relations in which there is a collection of ordered pairs.
But the difference lies in the fact that in function for each value of x there is unique value of y.

Difference between Relations and Function

Following is the illustration of a relation
You can see that in relations for a value of x , there can be multiple value of y

relations and functions concept ncert

And now observe the illustration of function
You can see that for every value of x there is unique value of y

relations and functions for Grade 11

Examples of Function and concept of Image

Difference between relations and functions

From the above figure we can see that:
f (1) = 4, which means for x=1, the image is 4
Similarly f (2) = 8, which means for x = 2, the image is 8

For every value of x, there is a unique image, so the above illustration is a function

Difference between relations and functions

For the above image:
f (4) = No image
f (3) = 6 and 1 { there are two images}

The above illustration is not a function because:
1. There is no image for x= 4
2. There are multiple images for x=3

what is the difference between relations and functions

From the above image we can say that:
f (5) = 6
Also for f(2) and f(3), the image is 6
The above illustration is function because there are only one value for every value of x

Representation of Function

If f is a function from A to B and (a, b) ∈ f, then f (a) = b,
The function f from A to B is denoted by f : A –> B

Let us understand the representation with the help of example
Let there is a set A such that
A = {1, 2, 3, 4}
And the function from A to B, f : A –> B is represented such that y = 2x, find the set B?

A = {1, 2, 3, 4}
Putting all these values as x in equation, y = 2x we get
B = {2, 4, 6, 8}

Important Function and their Graph

Identity Function

The function can be defined as y = f(x) = x
The set can be represented as P = {(1,1), (2, 2), (3,3), (4,4),(5,5) …..}

The graphical representation of this function is as follows

examples of function

Constant Function

This function can be represented as y = f (x) = c
Means for any value of x, the value of y is constant

Let us understand its graph of function y = f (x) = 3

what are function in Math

Modulus Function

The Modulus function is defined as:
f(x) = |x|

what is function in math

This function is like
a. if you put positive value of x, you will get the same positive value in return
For Example => (1, 1), (2, 2), (3, 3), (4, 4)

b. If you put negative value of x, you will get positive value of same number
For Example => (-1, 1), (-2, 2), (-3, 3), (-4, 4), (-5, 5)

So the final relation would be
R = { …….. (-3, 3), (-2, 2), (-1, 1), (0, 0), (1, 1), (2, 2), (3, 3), …..}

what are functions

Signum function

The signum function is defined as:

what are functions

The function is like
a. If value of x is greater than 0, then the value of y = 1
b. If value of x is equal to 0, then value of y = 0
c. If value of x is less than 0, then value of y = -1

The graph of the function is represented as follows

what is the difference between relation and funstions

I hope you have now understood the difference between relations and functions. This concept is extremely important as it sets the base to understand other advanced chapters.
Relation and Function is not only important for Grade 11 but the concept will be used in your higher education degree or research studies. So if you have not understood the concept well, i will strongly suggest to read the post again.

Leave a Comment

Your email address will not be published. Required fields are marked *

You cannot copy content of this page