   Chapter 11.2, Problem 18ES ### 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 11.2.7(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 ) ) . then g ( n ) is Θ ( f ( n ) ) .

To determine

To prove:

If f(n) is Θg(n), then g(n) is Θf(n) when f and g are real-valued functions defined on the same non-negative integers.

Explanation

Given information:

f and g are real-valued functions defined on same non-negative integers and f(n)0,g(n)0 for all nr and r is any real number greater than zero.

Formula used:

Let f and g be real valued functions defined on the same nonnegative integers, with g(n)0 for every integer nr, where r is positive real number.

Then,

f is of order g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and kr such that

Ag(n)f(n)Bg(n) for every integer nk.

Proof:

If f(n) is Θg(n), according to the definition of the Θ notation, there exist A,Bandk positive real numbers with

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