(a) Explanation Proof: For two functions f and g: N → R , we say that f = O ( g ) , if there is an integer n 0 and a positive real number c such that | f ( n ) | ≤ c | g ( n ) | for any n ≥ n 0 . Also, we say that f has a smaller order than g and write f ≺ g if and only if f = O ( g ) but g ≠ O ( f ) . We know prove that, 2 n ≺ n ! Now consider 2 n = 2 × 2 × 2 × ... × ( n t i m e s ) And n ! = 1 × 2 × 3 × ... × n Thus we can see that n ! will outgrow 2 n as soon as n becomes 4. Hence, 2 n < n ! for all n ≥ 4 Taking, c = 1 and n 0 = 4 , we have 2 n = O ( n ! ) ...... (b) To determine To prove: The b n ≺ n ! f o r a n y b > 1.

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

3rd Edition

ISBN: 9780134689555

Author: Edgar Goodaire, Michael Parmenter

Publisher: PEARSON

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Sequence and Series

A sequence is defined as a collection of things. Series is defined to sum the things one by one in the sequence. It was invented by German mathematician Carl Friedrich Gauss.

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Chapter 8.2, Problem 13E

(a)

To determine

To prove: The 2n≺n!.

(b)

To determine

To prove: The bn≺n!for any b>1.

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Chapter 8.2, Problem 12E

Chapter 8.2, Problem 14E

Chapter 8 Solutions

Discrete Mathematics with Graph Theory (Classic Version) (3rd Edition) (Pearson Modern Classics for Advanced Mathematics Series)

Ch. 8.1 - Prob. 1TFQ Ch. 8.1 - Prob. 2TFQ Ch. 8.1 - Prob. 3TFQ Ch. 8.1 - Prob. 4TFQ Ch. 8.1 - Prob. 5TFQ Ch. 8.1 - Prob. 6TFQ Ch. 8.1 - Prob. 7TFQ Ch. 8.1 - Prob. 8TFQ Ch. 8.1 - Prob. 9TFQ Ch. 8.1 - Prob. 10TFQ

Ch. 8.1 - Prob. 1E Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - Prob. 7E Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - Prob. 13E Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - Prob. 19E Ch. 8.1 - Prob. 20E Ch. 8.2 - Prob. 1TFQ Ch. 8.2 - Prob. 2TFQ Ch. 8.2 - Prob. 3TFQ Ch. 8.2 - Prob. 4TFQ Ch. 8.2 - Prob. 5TFQ Ch. 8.2 - Prob. 6TFQ Ch. 8.2 - Prob. 7TFQ Ch. 8.2 - Prob. 8TFQ Ch. 8.2 - Prob. 9TFQ Ch. 8.2 - Prob. 10TFQ Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - 4. Find an algorithm for finding the smallest...Ch. 8.2 - Prob. 5E Ch. 8.2 - 6. (a) [BB] Justify the statement made in...Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - Prob. 19E Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - The Russian peasant method is used to multiply two...Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.3 - Prob. 1TFQ Ch. 8.3 - Prob. 2TFQ Ch. 8.3 - (Answers can be found in the back of the book.)...Ch. 8.3 - Prob. 4TFQ Ch. 8.3 - Prob. 5TFQ Ch. 8.3 - (Answers can be found in the back of the book.) 6....Ch. 8.3 - Prob. 7TFQ Ch. 8.3 - Prob. 8TFQ Ch. 8.3 - Prob. 9TFQ Ch. 8.3 - Prob. 10TFQ Ch. 8.3 - Prob. 1E Ch. 8.3 - Prob. 2E Ch. 8.3 - Describe a ternary search algorithm, which...Ch. 8.3 - Prob. 4E Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - Prob. 7E Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - Prob. 13E Ch. 8.3 - Prob. 14E Ch. 8.3 - Prob. 15E Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - [BB] Show the steps involved in the application of...Ch. 8.3 - Prob. 19E Ch. 8.3 - The Binary search Algorithm we have presented...Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - Prob. 24E Ch. 8.3 - Prob. 25E Ch. 8.3 - Prob. 26E Ch. 8.4 - (Answers can be found in the back of the book.) 1....Ch. 8.4 - Prob. 2TFQ Ch. 8.4 - Prob. 3TFQ Ch. 8.4 - Prob. 4TFQ Ch. 8.4 - Prob. 5TFQ Ch. 8.4 - Prob. 6TFQ Ch. 8.4 - Prob. 7TFQ Ch. 8.4 - Prob. 8TFQ Ch. 8.4 - Prob. 9TFQ Ch. 8.4 - Prob. 10TFQ Ch. 8.4 - Prob. 1E Ch. 8.4 - Use the procedure outlined in this section to list...Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - 8. (a) List, in the lexicographic order, the...Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8 - Describe how Horners Algorithm evaluates f(x) when...Ch. 8 - Prob. 2RE Ch. 8 - 3. Let be an integer, let , and let be a subset of...Ch. 8 - Suppose we want an algorithm that, for an input of...Ch. 8 - Prob. 5RE Ch. 8 - Prob. 6RE Ch. 8 - Prob. 7RE Ch. 8 - Prob. 8RE Ch. 8 - (Requires a little knowledge of calculus) Show...Ch. 8 - Prob. 10RE Ch. 8 - Prob. 11RE Ch. 8 - 12. Sort the list 9,-3,1,0,-4,5,3 into increasing...Ch. 8 - 13. In the lexicographic ordering of all...Ch. 8 - Prob. 14RE

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To prove: The 2 n ≺ n ! .