f: X → Y
Function f is oneone if every element has a unique image,
i.e.
when f(x _{ 1 } ) = f(x _{ 2 } )
⇒ x _{ 1 } = x _{ 2 }
Otherwise the function is manyone.
How to check if function is oneone  Method 1
In this method, we check for each and every element manually if it has unique image
Check whether the following are oneone ?
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Element 
Image 
1 
a 
2 
b 
3 
c 
4 
d 
Since every element has a unique image,
it is oneone
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Element 
Image 
1 
b 
2 
c 
3 
d 
4 
a 
Since every element has a unique image,
it is oneone
Element 
Image 
1 
a 
2 
a 
3 
c 
4 
d 
Since 1 and 2 has same image,
it is not oneone
Element 
Image 
1 
a 
2 
b 
3 
c 
4 
d 
Since every element has a unique image,
it is oneone
How to check if function is oneone  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 oneone?
 Calculate f(x _{ 1 } )
 Calculate f(x _{ 2 } )
 Put f(x _{ 1 } ) = f(x _{ 2 } )
If x _{ 1 } = x _{ 2 } , then it is oneone.
Otherwise, manyone
Let’s take some examples
f: R → R
f(x) = x
Is f oneone?
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 oneone.
Otherwise, manyone
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 oneone.
ea
f: R → R
f(x) = 1
Is f oneone?
a
Since
f is not oneone.
ea