BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Solutions

Chapter
Section
BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
1 views

Suppose P ( n ) = a m n m + a m 1 n m 1 + + a 2 n 2 + a 1 n + a 0 , where all the coefficients

where all the coefficients a 0 , a 1 , , a m , are real numbers and a m > 0 .
a. Prove that P ( n ) is Ω ( n m ) by using the general procedure described in Example 11.2.4.
b. Prove that P ( n ) is O ( n m ) .
c. Justify the conclusion that P ( n ) is Θ ( n m )

To determine

(a)

Prove that P(n) is Ω(nm)

Explanation

Given information:

P(n)=amnm+am1nm1+...+a2n2+a1n+a0

Concept used:

f is of order at least g, written f(n) is Ω(g(n)), if and only if there exists positive real numbers A and ar such that,

Ag(n)f(n) for every integer na

Proof:

Let m be a non-negative integer, let P(n) be a polynomial of degree m, and suppose the coefficient am of nm is positive.

To find big-omega for P(n): Let a0

A=12am let’s find a as follows,

The coefficient of the highest power is am

Sum of absolute values of coefficients is |am1|+|am2|+...+|a2|+|a1|+|a0|. Thus,

A=12am and a=2am×(|am1|+|am2|+...+|a2|+|a1|+|a0|)

Requiring na means that,

n2am×(|am1|+|am2|+...+|a2|+|a1|+|a0|)

And multiplying both sides by nm12 gives,

nm(| a m1|+| a m2|+...+| a 2|+| a 1|+| a 0|)×n m1am1am(| a m1|n m1+| a m2|n m1+...+| a 2|n m1+| a 1|n m1+| a 0|n m1)1am(| a m1|n m1+| a m2|n m2+

To determine

(b)

Prove that P(n) is O(nm)

To determine

(c)

Justify the conclusion that P(n) is θ(nm)

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Chapter 11 Solutions

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