Chapter 4, Problem 17R

Explanation of Solution

Given:

It is given that if $\text{d}\left(\text{n}\right)\text{ is O}\left(\text{f}\left(\text{n}\right)\right)\text{ and e}\left(\text{n}\right)\text{ is O}\left(\text{g}\left(\text{n}\right)\right)$ then it is described that $\text{d}\left(\text{n}\right)-\text{e}\left(\text{n}\right)\text{ is not neccessary of O}\left(\text{f}\left(\text{n}\right)-\text{g}\left(\text{n}\right)\right)$.

Big-Oh notation:

In big-Oh notation, let “f” and “g” be functions from the integers or the real numbers to the real numbers. It means that $\text{f}\left(\text{x}\right)$ is “$\text{O}\left(\text{g}\left(\text{x}\right)\right)$” if there are constants “C” and “k” such that,

$\left|\text{f}\left(\text{x}\right)\right|\le \text{C}\left|\text{g}\left(\text{x}\right)\right|\text{ where }\left(\text{x > k}\right)$

Proof:

Assume that $\text{d}\left(\text{n}\right)$ is $\text{O}\left(\text{f}\left(\text{n}\right)\right)$, and $\text{e}\left(\text{n}\right)$ is $\text{O}\left(\text{g}\left(\text{n}\right)\right)$.

According to definition of big-oh notation, it is known that

$\text{d}\left(\text{n}\right)\le \text{ K f}\left(\text{n}\right)\text{for n }\ge \text{ N, and e}\left(\text{n}\right)\le \text{ L g}\left(\text{n}\right)\text{ for n }\ge \text{ M}$

Let us consider that $$