# 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.

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)

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

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

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

### 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

#### 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

### Questions

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.

Solution
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

(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

Solution
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}}

Solution
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

## 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

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

### Examples of Function and concept of Image

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

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

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

Example
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?

Solution:
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

#### 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

#### Modulus Function

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

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), …..}

#### Signum function

The signum function is defined as:

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

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.