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Math
Discrete Mathematics With Applications5th Edition

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Chapter 11.2, Problem 51ES
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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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Problem 4ES
Problem 5ES
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Problem 11ES
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Problem 13ES
Problem 14ES
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Problem 25ES
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Problem 28ES
Problem 29ES
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Problem 33ES
Problem 34ES
Problem 35ES
Problem 36ES
Problem 37ES
Problem 38ES
Problem 39ES
Problem 40ES
Problem 41ES
Problem 42ES
Problem 43ES
Problem 44ES
Problem 45ES
Problem 46ES
Problem 47ES
Problem 48ES
Problem 49ES
Problem 50ES
Problem 51ES
BuyFindarrow_forward

Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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Prove Theorem 11.2.9:
a. Let f 1 ,   f 2 , and g be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 ,   f 2 ( n ) ≥ 0 , and g ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g ( n ) ) and f 2 ( n ) is Θ ( g ( n ) ) , then ( f 1 ( n ) + f 2 ( n ) ) is Θ ( g ( n ) ) .

b. Let f 1 ,   f 2 ,   g 1 , and g 2 , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 ,   f 2 ( n ) ≥ 0 ,   g 1 ( n ) ≥ 0 , and g 2 ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g 1 ( n ) ) and f 2 ( n ) is Θ ( g 2 ( n ) ) , then ( f 1 ( n ) f 2 ( n ) ) is Θ ( g 1 ( n ) g 2 ( n ) ) .

c. Let f 1 ,   f 2 ,   g 1 , and g 2 , be real-valued functions defined on the same set of nonnegative integers, and suppose there is a positive real number r such that f 1 ( n ) ≥ 0 ,   f 2 ( n ) ≥ 0 ,   g 1 ( n ) ≥ 0 , and g 2 ( n ) ≥ 0 for every integer n ≥ r . If f 1 ( n ) is Θ ( g 1 ( n ) ) and f 2 ( n ) is Θ ( g 2 ( n ) ) and if there is a real number s so that g 1 ( n ) ≤ g 2 ( n ) for every integer n ≥ s , then ( f 1 ( n ) + f 2 ( n ) ) is Θ ( g 2 ( n ) ) .

To determine

(a)

To prove:

That for the real-valued functions f1, f2 and g, if f1(n) is Θ(g(n)) and f2(n) is Θ(g(n)) then (f1(n)+f2(n)) is Θ(g(n)).

Explanation

Given information:

The functions f1, f2 and g are defined on same set of non-negative integers and for every n≥r, f1(n)≥0, f2(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 n≥r, where r is positive real number.

Then,

f is of order at least g, written f(n) is Θ(g(n)), if and only if, there exist positive real numbers A,B and a≥r such that

Ag(n)≤f(n)≤Bg(n) for every integer n≥a.

Proof:

Using the definition of Θ− notation,

there exists four real numbers A1≥0,A2≥0,B1≥0 and B2≥0 such that A1g(n)≤f1(n)≤B1g(n) and A2g(n)≤f2(n)≤B2g(n) for every n≥r

To determine

(b)

To prove:

That for the real-valued functions f1, f2, g1 and g2, if f1(n) is Θ(g1(n)) and f2(n) is Θ(g2(n)) then f1(n)f2(n) is Θ(g1(n)g2(n)).

To determine

(c)

To prove:

That for the real-valued functions f1, f2, g1 and g2, if f1(n) is Θ(g1(n)) and f2(n) is Θ(g2(n)) then f1(n)+f2(n) is Θ(g2(n)) when g1(n)≤g2(n).

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Chapter 11 Solutions

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Sect-11.1 P-1TYSect-11.1 P-2TYSect-11.1 P-3TYSect-11.1 P-4TYSect-11.1 P-5TYSect-11.1 P-6TYSect-11.1 P-1ESSect-11.1 P-2ESSect-11.1 P-3ESSect-11.1 P-4ES
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
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