20. 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) both onto and one-to-one (but different from the identity function). d) neither one-to-one nor onto.

Step-by-step explanation:

a) To provide an example of a function N → N that is one-to-one but not onto.

Suppose  to be

Then;

Now, assuming

Then

The above function is said to be one-to-one

e.g

2 is not a perfect square, hence, it is not regarded as the image of any natural no.

As such, f is not onto.

We can thereby conclude that the function   is one-to-one but not onto

b)

be

It implies that the function is not one-to-one since there exist different natural no. having the same image.

So, for , there exists an image of 2n in N

i.e.

Hence, the function is onto

We thereby conclude that the function

c)

be  

So, if n, m is odd:

Then:

Likewise, if n, m is even:

Then;

The function is then said to be one-to-one.

However, For and is odd, there exists an image of that is even;

For and is even, there exists an image of that is odd;

where(; implies such that)

Hence, this function is said to be onto.

We can therefore conclude that the function is both onto and one-to-one.

d)

Here, to provide an example where the is neither one-to-one nor onto.

SO;

Let is defined to be

Then, since every integer has the same image as zero(0), the function is not one-to-one.

Similarly, the function is not onto since every positive integer is not an image of any natural number.

We, therefore conclude that, the function is neither one-to-one nor onto.

c

step-by-step explanation:

what