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Big-O, Big-Theta and big-Omega notation can be extended to

Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen ISBN: 9780073383095 37

Solution for problem 58E Chapter 3.2

Discrete Mathematics and Its Applications | 7th Edition

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Discrete Mathematics and Its Applications | 7th Edition | ISBN: 9780073383095 | Authors: Kenneth Rosen

Discrete Mathematics and Its Applications | 7th Edition

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Problem 58E

Problem 58E

Big-O, Big-Theta and big-Omega notation can be extended to functions in more than one variable. For example, the statement f (x, y) is O (g(x, y)) means that there exist constants C, k1 , and k2 such that whenever x > k1 and y > k2 .

(Requires calculus) Show that if b > 1 and c and d are positive, then (logbn)c is O(nd), but nd is not O((logbn)c).

Step-by-Step Solution:
Step 1 of 3

Solution:

Step1

Given that

We have to show that if b > 1 and c and d are positive, then (logbn)c is O(nd), but nd is not O((logbn)c).

Step2

 The statement f (x, y) is O (g(x, y)) means that there exist constants C, k1 , and k2 such that whenever x > k1 and y > k2 .

Let us assume that f and g is functions from the set of integers.

We can say that f(x) is O(g(x)) if there are constants C and k such that

Where >k.

The constants C and k are said to be witnesses to the relationship.

If b>1 means all n

                 

So, for C=1 and k=1

It is clear that  is

Step3

For nd is not O((logbn)c)

It is not possible because exponential function will always grow faster than the logarithmic function.

Step 2 of 3

Chapter 3.2, Problem 58E is Solved
Step 3 of 3

Textbook: Discrete Mathematics and Its Applications
Edition: 7
Author: Kenneth Rosen
ISBN: 9780073383095

Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. The full step-by-step solution to problem: 58E from chapter: 3.2 was answered by , our top Math solution expert on 06/21/17, 07:45AM. The answer to “Big-O, Big-Theta and big-Omega notation can be extended to functions in more than one variable. For example, the statement f (x, y) is O (g(x, y)) means that there exist constants C, k1 , and k2 such that whenever x > k1 and y > k2 . (Requires calculus) Show that if b > 1 and c and d are positive, then (logbn)c is O(nd), but nd is not O((logbn)c).” is broken down into a number of easy to follow steps, and 70 words. This full solution covers the following key subjects: Calculus, log, Positive, requires, show. This expansive textbook survival guide covers 101 chapters, and 4221 solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Since the solution to 58E from 3.2 chapter was answered, more than 304 students have viewed the full step-by-step answer.

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Big-O, Big-Theta and big-Omega notation can be extended to