Question

Give an example of two increasing functions$\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}$and$\mathit{g}\mathbf{\left(}\mathit{n}\mathbf{\right)}$from the set of positive integers to the set of positive integers such that neither$\mathit{f}\mathbf{\left(}\mathit{n}\mathbf{\right)}$is$\mathit{O}\mathbf{\left(}\mathit{g}\mathbf{\right(}\mathit{n}\mathbf{\left)}\mathbf{\right)}$nor$\mathit{g}\mathbf{\left(}\mathit{n}\mathbf{\right)}$is$\mathit{O}\mathbf{\left(}\mathit{f}\mathbf{\right(}\mathit{n}\mathbf{\left)}\mathbf{\right)}$.

Short answer

An example of two increasing functions are$f\left(n\right)=0$ ifn is odd and$f\left(n\right)=n^n$ ifn is even and$g\left(n\right)=n$

Step-by-step solution

Example of two increasing functions 

An example for the function$f\left(n\right)$ is

$f\left(n\right)=0$if is odd and

$f\left(n\right)=n^n$ if n is even

An example for the function $g\left(n\right)$is

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