Question

a) Define what it means for a function from the set of positive integers

to the set of positive integers to be one-to-one

b) Define what it means for a function from the set of positive integers to the set

of positive integers to be onto.

c) Give an example of a function from the set of positive integers to the set of

positive integers that is both one-to-one and onto.

d) Give an example of a function from the set of positive integers to the set of

positive integers that is one-to-one but not onto.

e) Give an example of a function from the set of positive integers to the set of

positive integers that is not one-to-one but is onto.

f) Give an example of a function from the set of positive integers to the set of

positive integers that is neither one-to-one nor onto.

Short answer

a) Function f is one-one if$\mathrm{f}\left(\mathrm{a}\right)=\mathrm{f}\left(\mathrm{b}\right)$ implies that $\mathrm{a}=\mathrm{b}$for all and in the domain.

b) Function$\mathrm{f}:\mathrm{A}\to \mathrm{B}$ is onto if for every element there exist an element$a\in A$ such that$\mathrm{f}\left(\mathrm{a}\right)=\mathrm{b}$

c) Required answer is $\mathrm{f}\left(\mathrm{n}\right)=\mathrm{n}$.

d) Required answer is $\mathrm{f}\left(\mathrm{n}\right)=2\mathrm{n}$

e) Required answer is$f\left(n\right)=\left\{\begin{array}{l}\left(\frac{n}{2}\right)\text{if}n\text{even}\\ \left(\frac{n+1}{2}\right)\text{if}n\text{odd}\end{array}\right.$

f) Required answer is $\mathrm{f}\left(\mathrm{n}\right)=1$.

Step-by-step solution

Step 1:Definition of funtion

A function$\mathbf{f}\mathbf{:}\mathbf{A}\mathbf{\to }\mathbf{B}$ has the property that each element of a has been mapped to exactly one element of B.

Step 2: One-One function

a) The function $f:A\to B$is one to one if and only if $f\left(a\right)=f\left(b\right)$implies that a=b for all a and b in the domain.

Onto function

b) The function $f:A\to B$is onto if and only if for every element$b\in B$ there exist an element$a\in A$ such that$f\left(a\right)=b$

Also, function f is onto if range of function is equal to co-domain of function.

One-One and onto function

c) We need to find a function from the positive integers to the positive integers$\mathrm{f}:{\mathrm{Z}}^{+}\to {\mathrm{Z}}^{+}$

such that function is both one –one and onto.

For example,

$f\left(n\right)=n$

is one to one let$\mathrm{f}\left(\mathrm{a}\right)=\mathrm{f}\left(\mathrm{b}\right)\Rightarrow \mathrm{a}=\mathrm{b}$

Is onto since for all $b\in Z^{+}$, there exists$a\in Z^{+}:f\left(a\right)=b$

One-One and but not onto function

(d) We need to find a function f from the set of positive integers to the set of positive integers$f:Z^{+}\to Z^{+}$

such that function is one –one but not onto.

For example,

$f\left(n\right)=2n$

f is one to one let $\mathrm{f}\left(\mathrm{a}\right)=\mathrm{f}\left(\mathrm{b}\right)$,by definition of $f:2a=2b$. Dividing each side of the equation by 2, we get a=b

is not onto: let $\mathrm{b}\in {\mathrm{z}}^{+}$, then we $a=\frac{b}{2}$is the only value with the property$f\left(a\right)=f\left(\frac{b}{2}\right)=2\left(\frac{b}{2}\right)=b$

However, if b is odd, then $a=\frac{b}{2}$is not a positive integer$(a\notin Z^{+})$ and thus f is not onto.

Not one-one but onto function

e)

We need to find a function f from the set of positive integers to the set of positive integers

$f:Z^{+}\to Z^{+}$

such that function is not one –one but onto.

For example,

$f\left(n\right)=\left\{\begin{array}{l}\left(\frac{n}{2}\right)\text{if}n\text{even}\\ \left(\frac{n+1}{2}\right)\text{if}n\text{odd}\end{array}\right.$

Thus, the image of and is then, the image of and is , the image of and is , the image of and is , and so on.

is not one to one because n=1,2 have the same image.

is onto since range is equal to co-domain.

Neither one-One and onto function

f)

We need to find a function f from the set of positive integers to the set of positive integers such that

$f:Z^{+}\to Z^{+}$

And f is neither one-one nor onto.

For example, let

$f\left(n\right)=1$

is not one to one because all positive integers have the same image.

is not onto because all positive integers (except for 1) are not in the range.