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

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### 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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