Chapter 11.2, Problem 50ES

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

Chapter
Section

### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# Prove Theorem 11.2.8: a. Let f 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 ( n ) ≥ 0 and g ( n ) > 0 for every integer n ≥ r . If f ( n ) is Ω ( g ( n ) ) and c is any positive real number, then c f ( n ) is Ω ( g ( n ) ) . b. Let f 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 ( n ) ≥ 0 and g ( n ) ≥ 0 for every integer n ≥ r . If f ( n ) is O ( g ( n ) ) and c is any positive real number, then c f ( n ) is O ( g ( n ) ) . c. Let f 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 ( n ) ≥ 0 and g ( n ) ≥ 0 for every integer n ≥ r . If f ( n ) is Θ ( g ( n ) ) and c is any positive real number, then c f ( n ) is Θ ( g ( n ) ) .

To determine

(a)

To prove:

That for the real-valued function f and g, if f(n) is Ω(g(n)), then cf(n) is Ω(g(n)) for any real number c.

Explanation

Given information:

The functions f and g are defined on same set of non-negative integers and for every nr, f(n)0 and g(n)0 where r is a positive real number. Also c is any 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 Ω(g(n)), if and only if, there exist positive real number A and ar such that

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

Proof:

Using the definition of Ω notation,

there exists a real number A0 such that Ag(n)f(n) for every nr

To determine

(b)

To prove:

That for the real-valued function f and g, if f(n) is O(g(n)), then cf(n) is O(g(n)) for any real number c.

To determine

(c)

To prove:

That for the real-valued function f and g, if f(n) is θ(g(n)), then cf(n) is θ(g(n)) for any real number c.

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