Give an example of a function from N to N that is:. a) one-to-one but not onto
b) onto but not one-to-one
c) neither one-to-one nor onto

Let f be a function

a) f(n) = n²

b) f(n) = n/2

c) f(n) = 0

Explanation:

a) f(n) = n²

This function is one-to-one function because the square of two different or distinct natural numbers cannot be equal.

Let a and b are two elements both belong to N i.e. a ∈ N and b ∈ N. Then:

                                f(a) = f(b) ⇒ a² = b² ⇒ a = b

The function f(n)= n² is not an onto function because not every natural number is a square of a natural number. This means that there is no other natural number that can be squared to result in that natural number.  For example 2 is a natural numbers but not a perfect square and also 24 is a natural number but not a perfect square.

b) f(n) = n/2

The above function example is an onto function because every natural number, lets say n is a natural number that belongs to N, is the image of 2n. For example:

                                f(2n) = [2n/2] = n

The above function is not one-to-one function because there are certain different natural numbers that have the same value or image. For example:

When the value of n=1, then    

                                   n/2 = [1/2] = [0.5] = 1

When the value of n=2 then

                                    n/2 = [2/2] = [1] = 1

c) f(n) = 0

The above function is neither one-to-one nor onto. In order to depict that a function is not one-to-one there should be two elements in N having same image and the above example is not one to one because every integer has the same image.  The above function example is also not an onto function because every positive integer is not an image of any natural number.

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