Check sibling questions

f: X → Y

Function f is one-one if every element has a unique image,

i.e.

when f(x 1 ) = f(x 2 )

⇒ x 1 = x 2

Otherwise the function is many-one.

 

How to check if function is one-one - Method 1

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

 

Check whether the following are one-one ?

1.jpg

Element

Image

1

a

2

b

3

c

4

d

Since every element has a unique image,

it is one-one     

 


One One function - Part 2


Element

Image

1

b

2

c

3

d

4

a

Since every element has a unique image,

it is one-one    


One One function - Part 3

Element

Image

1

a

2

a

3

c

4

d

Since 1 and 2 has same image,

it is not one-one


One One function - Part 4

Element

Image

1

a

2

b

3

c

4

d

Since every element has a unique image,

it is one-one     

 


How to check if function is one-one -  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 one-one?

  1. Calculate f(x 1 )   
  2. Calculate f(x 2 )   
  3. Put f(x 1 ) = f(x 2 )  

If x 1 = x 2 , then it is one-one.

Otherwise, many-one

 

Let’s take some examples

f: R R

f(x) = x

Is f one-one?

-a-

We follow the steps

  1. Calculate f(x 1 )
  2. Calculate f(x 2 )
  3. Put f(x 1 ) = f(x 2 ),
  4. If x 1 = x 2 , then it is one-one.
    Otherwise, many-one

 

f(x 1 ) = x 1

f(x 2 ) = x 2

Putting f(x 1 ) = f(x 2 )

x 1 = x 2

 

Since x 1 = x 2 ,

f is one-one.

-ea-

 

f: R R

f(x) = 1

Is f one-one?

-a-

Since

One One function - Part 5

f is not one-one.

-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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.