Chapter 2, Problem 2E

Program Plan Intro

Big O definition:

The Big O definitions for “f₁(n)” and “f₂(n)” are shown below:

• The function “f₁(n)” is O(g₁(n) if positive numbers “c₁” and “N₁” exists such that $f_1\left(n\right)\le c_1{g}_1\left(n\right)$ for all $n\ge N_1$.
• The function “f₂(n)” is O(g₂(n) if positive numbers “c₂” and “N₂” exists such that $f_2\left(n\right)\le c_2{g}_2\left(n\right)$ for all $n\ge N_2$.

Explanation of Solution

(b) $O\left(g_1\left(n\right)+g_2\left(n\right)\right)$ is $O\left(g_2\left(n\right)\right)$

• Let $g_1\left(n\right)\le g_2\left(n\right)$ and $c=max\left(c_1,c_2\right)$.
• Multiply both sides of equation by constant “c” to obtain $c{g}_1\left(n\right)\le c{g}_2\left(n\right)$

Explanation of Solution

(c) $f_1\left(n\right).f_2\left(n\right)$ is $O\left(g_1\left(n\right).g_2\left(n\right)\right)$

• The product rule, $f_1\left(n\right).f_2\left(n\right)$ is $O(g_1\left(n\right)$

Explanation of Solution

(d) $O(cg(n))$ is $O(g(n))$

• The statement $O(cg(n))$ is $O(g(n))$ means that any function “f” which is $O(cg)$ is also $O(g)$

Explanation of Solution

(e) “c” is O(1)

• A constant “c” is O(1) if positive numbers “c₁” and “N” exists such that $c\le c_1.1$ for all $n\ge N$