   Chapter 11.2, Problem 19ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove Theorem 11.2.1(b): If f and g are real-valued functions defined on the same set of nonnegative integers and if f ( n ) ≥ 0 and g ( n ) ≥ 0 for every integer n ≥ r , where r is a positive real number, then if f ( n ) is Θ ( g ( n ) ) if, and only if f ( n ) is Ω ( g ( n ) ) and f ( n ) is O ( g ( n ) ) .

To determine

The proof of the given statement.

Explanation

Given information:

The given statement is,

“Two real valued functions f and g are defined on the same set of non-negative integers and if f(n)0 and g(n)0 for every integer nr, where r is a positive real number, then f(n) is Θ(g(n)) if and only if f(n) is Ω(g(n)) and f(n) is Ο(g(n)) ”.

Proof:

Let’s assume f(n)=

Ω(g(n)), then according to the definition of Ω ,

A|g(n)||f(n)|

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