   Chapter 11.2, Problem 48ES ### 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 11.2.6(b): If f and g are real-valued functions defined on the same set of nonnegative integers, and if there is a positive real number r such that f ( n ) ≥ 0 and g ( n ) ≥ 0 for every integer n r, and if g ( n ) is O ( f ( n ) ) , then f ( n ) is Ω ( g ( n ) ) .

To determine

To prove:

That for f and g real valued functions defined on same set of non-negative integers, if g(n) is O(f(n)), then f(n) is Ω(g(n)).

Explanation

Given information:

The functions f and g are defined on same set of non-negative integers and for every nr, f(n)0 and g(n)0 where r is a positive real number.

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,

1. f is of order at least g, written f(n) is Ω(g(n)), if and only if, there exist positive real number A and ar such that
2. Ag(n)f(n) for every integer na.

3. f is of order at most g, written f(n) is O(g(n)), if and only if, there exist positive real number B and br such that
4. 0f(n)Bg(n) for every integer nb

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