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 ?
|
Element |
Image |
|
1 |
a |
|
2 |
b |
|
3 |
c |
|
4 |
d |
Since every element has a unique image,
it is one-one
|
Element |
Image |
|
1 |
b |
|
2 |
c |
|
3 |
d |
|
4 |
a |
Since every element has a unique image,
it is one-one
|
Element |
Image |
|
1 |
a |
|
2 |
a |
|
3 |
c |
|
4 |
d |
Since 1 and 2 has same image,
it is not one-one
|
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?
- Calculate f(x 1 )
- Calculate f(x 2 )
- 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
- Calculate f(x 1 )
- Calculate f(x 2 )
- Put f(x 1 ) = f(x 2 ),
-
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
f is not one-one.
-ea-
