# Prove Theorem 11.2.9: a . Let f 1 , f 2 , and g be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 , f 2 ( n ) ≥ 0 , and g ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g ( n ) ) and f 2 ( n ) is Θ ( g ( n ) ) , then ( f 1 ( n ) + f 2 ( n ) ) is Θ ( g ( n ) ) . b. Let f 1 , f 2 , g 1 , and g 2 , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 , f 2 ( n ) ≥ 0 , g 1 ( n ) ≥ 0 , and g 2 ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g 1 ( n ) ) and f 2 ( n ) is Θ ( g 2 ( n ) ) , then ( f 1 ( n ) f 2 ( n ) ) is Θ ( g 1 ( n ) g 2 ( n ) ) . c. Let f 1 , f 2 , g 1 , and g 2 , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 , f 2 ( n ) ≥ 0 , g 1 ( n ) ≥ 0 , and g 2 ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g 1 ( n ) ) and f 2 ( n ) is Θ ( g 2 ( n ) ) and if there is a real number s so that g 1 ( n ) ≤ g 2 ( n ) for every integer n ≥ s , then ( f 1 ( n ) + f 2 ( n ) ) is Θ ( g 2 ( n ) ) .

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
Publisher: Cengage Learning,
ISBN: 9781337694193

#### Solutions

Chapter
Section
Chapter 11.2, Problem 51ES
Textbook Problem

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