   Chapter 11.2, Problem 40ES ### 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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# a. Prove: If c is a positive real number and if f is a real-valued function defined on a set of nonnegative integers with f ( n ) ≥ 0 for every integer n greater than or equal to some positive real number, then c f ( n ) is Θ ( f ( n ) ) .b. Use part (a) to show that 3n is Θ ( n ) .

To determine

(a)

To prove:

c(f(n)) is Θ(f(n)), if c is a positive real number and if f is a real valued function defined on a set of non-negative integers with f(n)0 for every integer n greater than or equal to some positive real number.

Explanation

Given:

If c is a positive real number and if f is a real valued function defined on a set of non-negative integers with f(n)0 for every integer n greater than or equal to some positive real number, then c(f(n)) is Θ(f(n)).

Calculation:

Suppose that f is a real valued function defined on a set of non-negative integers with f(n)0 for every integer n greater than or equal to some positive real number.

Let nk.

By definition of big-O notation,

f(n) is O(f(n))

Therefore, for there exist real number B ,

0f(n)Bg(n) for every integer nk ; where g(n

To determine

(b)

To show that 3n is Θ(n) using part (a).

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