To prove one-one & onto (injective, surjective, bijective)

Chapter 1 Class 12 Relation and Functions
Concept wise

f: X → Y

Function f is onto if every element of set Y has a pre-image in set X

i.e.

For every y ∈ Y,

there is x ∈ X

such that f(x) = y

### How to check if function is onto - Method 1

In this method, we check for each and every element manually if it has unique image

Check whether the following are onto?

Since all elements of set B has a pre-image in set A,

it is onto

Since all elements of set B has a pre-image in set A,

it is onto

Since element b has no pre-image,

it is not onto

Since element e has no pre-image,

it is not onto

### How to check if function is onto -  Method 2

This method is used if there are large numbers

Example:

f : N N   (There are infinite number of natural numbers)

f : R R   (There are infinite number of real numbers )

f : Z Z    (There are infinite number of integers)

Steps :

How to check onto?

1. Put y =  f(x)
2. Find x in terms of y.

If x ∈ X, then f is onto

Let’s take some examples

f: R R

f(x) = x

Is f onto?

-a-

1. Put y = f(x)
2. Find x in terms of y.

If x ∈ X, then f is onto

y = f(x)

y = x

∴ x = y

Since y ∈ R

x  = y also belongs to R

i.e. x ∈ R

∴ f is onto

-ea-

f: R → R

f(x) = 1

Is f onto?

-a-

f(x) = 1

∴ y = 1

So, value of y will always be 1

∴ There is no value x where y = 2

⇒ 2 does not have a pre-image in X

∴ f is not onto

-ea-

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#### Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.