   Chapter 11.2, Problem 49ES ### 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(a): If f is a real-valued function defined on a set of nonnegative integers

To determine

To prove:

That for every nr, f(n) is Θ(f(n)).

Explanation

Given information:

f(n) is a real-valued function where n is any non-negative integer and there exists a real number r such that for every nr, f(n)0.

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 ka.

Proof:

Suppose there exists an integer A such that A1 ,

Then, it is clear that 0f(n)Af(n) for every nr

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