Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

ISBN: 9781337694193

Textbook Problem

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a. Show that for any integer n ≥ 1 , 0 ≤ 23 n 4 + 8 n 2 + 4 n ≤ 35 n 4 .
b. Show that for any integer n ≥ 1 , 23 n 4 ≤ 23 n 4 + 8 n 2 + 4 n .
c. Sketch a graph to illustrate the result of parts (a) and (b).
d. Use the O- and Ω -notations to express the results of parts (a) and (b).
e. What can you deduce about the order of 23 n 4 + 8 n 2 + 4 n ?

To determine

(a)

To prove:

The function 23n4+8n2+4n is greater than or equal 0 and less than or equal 35n4 for n≥1.

Explanation

Given information:

The functions 23n4+8n2+4n and 35n4 are defined on n and all n are positive integers where n≥1.

Proof:

Let f(n)=23n4+8n2+4n.

It is clear that for any integer n≥0, f(n) is greater than or equal zero.

As an example,

let n=1,f(n=1)=23(1)4+8(1)2+4(1)=23+8+4=35

Hence, f(1)≥0

Therefore, f(n)≥0 for any integer n≥0.

Also, when n=1, f(1)=35 as we proved previously.

Then,

f(n=1)=35=35(1)=35(1)4f(n=1)=35(n)4

Let’s assume that f(p)≤35p4 for any integer n=p where n>1.

Hence,

f(p)=23p4+8p2+4p≤35p4

when n=p+1 ,

f(p+1)=23(p+1)4+8(p+1)2+4(p+1)=23(p4+4p3+6p2+4p+1)+8(p2+2p+1)+4p+4=(23p4+8p2+4p)+(92p3+138p2+112p+35)f

To determine

(b)

To prove:

The function 23n4+8n2+4n is greater than or equal 23n4 for n≥1.

To determine

(c)

To graph:

An illustration about the variation of the functions in previous part(a.) and party(b.)

( 0≤23n4+8n2+4n≤35n4 and 23n4+8n2+4n≥23n4 )

To determine

(d)

The relationship between the functions 23n4+8n2+4n, 35n4 and 23n4 in the notations of O− and Ω−.

To determine

(e)

The order of the functions 23n4+8n2+4n

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Chapter 11 Solutions

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Sect-11.1 P-1TY Sect-11.1 P-2TY Sect-11.1 P-3TY Sect-11.1 P-4TY Sect-11.1 P-5TY Sect-11.1 P-6TY Sect-11.1 P-1ES Sect-11.1 P-2ES Sect-11.1 P-3ES Sect-11.1 P-4ES

Sect-11.1 P-5ES Sect-11.1 P-6ES Sect-11.1 P-7ES Sect-11.1 P-8ES Sect-11.1 P-9ES Sect-11.1 P-10ES Sect-11.1 P-11ES Sect-11.1 P-12ES Sect-11.1 P-13ES Sect-11.1 P-14ES Sect-11.1 P-15ES Sect-11.1 P-16ES Sect-11.1 P-17ES Sect-11.1 P-18ES Sect-11.1 P-19ES Sect-11.1 P-20ES Sect-11.1 P-21ES Sect-11.1 P-22ES Sect-11.1 P-23ES Sect-11.1 P-24ES Sect-11.1 P-25ES Sect-11.1 P-26ES Sect-11.1 P-27ES Sect-11.1 P-28ES Sect-11.2 P-1TY Sect-11.2 P-2TY Sect-11.2 P-3TY Sect-11.2 P-4TY Sect-11.2 P-5TY Sect-11.2 P-6TY Sect-11.2 P-1ES Sect-11.2 P-2ES Sect-11.2 P-3ES Sect-11.2 P-4ES Sect-11.2 P-5ES Sect-11.2 P-6ES Sect-11.2 P-7ES Sect-11.2 P-8ES Sect-11.2 P-9ES Sect-11.2 P-10ES Sect-11.2 P-11ES Sect-11.2 P-12ES Sect-11.2 P-13ES Sect-11.2 P-14ES Sect-11.2 P-15ES Sect-11.2 P-16ES Sect-11.2 P-17ES Sect-11.2 P-18ES Sect-11.2 P-19ES Sect-11.2 P-20ES Sect-11.2 P-21ES Sect-11.2 P-22ES Sect-11.2 P-23ES Sect-11.2 P-24ES Sect-11.2 P-25ES Sect-11.2 P-26ES Sect-11.2 P-27ES Sect-11.2 P-28ES Sect-11.2 P-29ES Sect-11.2 P-30ES Sect-11.2 P-31ES Sect-11.2 P-32ES Sect-11.2 P-33ES Sect-11.2 P-34ES Sect-11.2 P-35ES Sect-11.2 P-36ES Sect-11.2 P-37ES Sect-11.2 P-38ES Sect-11.2 P-39ES Sect-11.2 P-40ES Sect-11.2 P-41ES Sect-11.2 P-42ES Sect-11.2 P-43ES Sect-11.2 P-44ES Sect-11.2 P-45ES Sect-11.2 P-46ES Sect-11.2 P-47ES Sect-11.2 P-48ES Sect-11.2 P-49ES Sect-11.2 P-50ES Sect-11.2 P-51ES Sect-11.3 P-1TY Sect-11.3 P-2TY Sect-11.3 P-3TY Sect-11.3 P-1ES Sect-11.3 P-2ES Sect-11.3 P-3ES Sect-11.3 P-4ES Sect-11.3 P-5ES Sect-11.3 P-6ES Sect-11.3 P-7ES Sect-11.3 P-8ES Sect-11.3 P-9ES Sect-11.3 P-10ES Sect-11.3 P-11ES Sect-11.3 P-12ES Sect-11.3 P-13ES Sect-11.3 P-14ES Sect-11.3 P-15ES Sect-11.3 P-16ES Sect-11.3 P-17ES Sect-11.3 P-18ES Sect-11.3 P-19ES Sect-11.3 P-20ES Sect-11.3 P-21ES Sect-11.3 P-22ES Sect-11.3 P-23ES Sect-11.3 P-24ES Sect-11.3 P-25ES Sect-11.3 P-26ES Sect-11.3 P-27ES Sect-11.3 P-28ES Sect-11.3 P-29ES Sect-11.3 P-30ES Sect-11.3 P-31ES Sect-11.3 P-32ES Sect-11.3 P-33ES Sect-11.3 P-34ES Sect-11.3 P-35ES Sect-11.3 P-36ES Sect-11.3 P-37ES Sect-11.3 P-38ES Sect-11.3 P-39ES Sect-11.3 P-40ES Sect-11.3 P-41ES Sect-11.3 P-42ES Sect-11.3 P-43ES Sect-11.4 P-1TY Sect-11.4 P-2TY Sect-11.4 P-3TY Sect-11.4 P-4TY Sect-11.4 P-5TY Sect-11.4 P-1ES Sect-11.4 P-2ES Sect-11.4 P-3ES Sect-11.4 P-4ES Sect-11.4 P-5ES Sect-11.4 P-6ES Sect-11.4 P-7ES Sect-11.4 P-8ES Sect-11.4 P-9ES Sect-11.4 P-10ES Sect-11.4 P-11ES Sect-11.4 P-12ES Sect-11.4 P-13ES Sect-11.4 P-14ES Sect-11.4 P-15ES Sect-11.4 P-16ES Sect-11.4 P-17ES Sect-11.4 P-18ES Sect-11.4 P-19ES Sect-11.4 P-20ES Sect-11.4 P-21ES Sect-11.4 P-22ES Sect-11.4 P-23ES Sect-11.4 P-24ES Sect-11.4 P-25ES Sect-11.4 P-26ES Sect-11.4 P-27ES Sect-11.4 P-28ES Sect-11.4 P-29ES Sect-11.4 P-30ES Sect-11.4 P-31ES Sect-11.4 P-32ES Sect-11.4 P-33ES Sect-11.4 P-34ES Sect-11.4 P-35ES Sect-11.4 P-36ES Sect-11.4 P-37ES Sect-11.4 P-38ES Sect-11.4 P-39ES Sect-11.4 P-40ES Sect-11.4 P-41ES Sect-11.4 P-42ES Sect-11.4 P-43ES Sect-11.4 P-44ES Sect-11.4 P-45ES Sect-11.4 P-46ES Sect-11.4 P-47ES Sect-11.4 P-48ES Sect-11.4 P-49ES Sect-11.4 P-50ES Sect-11.4 P-51ES Sect-11.5 P-1TY Sect-11.5 P-2TY Sect-11.5 P-3TY Sect-11.5 P-4TY Sect-11.5 P-5TY Sect-11.5 P-1ES Sect-11.5 P-2ES Sect-11.5 P-3ES Sect-11.5 P-4ES Sect-11.5 P-5ES Sect-11.5 P-6ES Sect-11.5 P-7ES Sect-11.5 P-8ES Sect-11.5 P-9ES Sect-11.5 P-10ES Sect-11.5 P-11ES Sect-11.5 P-12ES Sect-11.5 P-13ES Sect-11.5 P-14ES Sect-11.5 P-15ES Sect-11.5 P-16ES Sect-11.5 P-17ES Sect-11.5 P-18ES Sect-11.5 P-19ES Sect-11.5 P-20ES Sect-11.5 P-21ES Sect-11.5 P-22ES Sect-11.5 P-23ES Sect-11.5 P-24ES Sect-11.5 P-25ES Sect-11.5 P-26ES

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