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Discrete Mathematics With Applicat...

5th Edition
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ISBN: 9781337694193

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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 nr, 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 nr, 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 ar such that

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

Proof:

Using the definition of Θ notation,

there exists four real numbers A10,A20,B10 and B20 such that A1g(n)f1(n)B1g(n) and A2g(n)f2(n)B2g(n) for every nr

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-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