   Chapter 11.2, Problem 13ES ### 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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# Use the definition of Θ -notation to show that 5 n 3 + 65 n + 30 is Θ ( n 3 ) .

To determine

To show that 5n3+65n+30 is Θ(n3).

Explanation

Given information:

5n3+65n+30.

Concept used:

Let f and g be real-valued functions defined on the same set of nonnegative integers, with g(n)0 for every integer nr, where r is a positive real number. Then f is of order g, written f ( n ) is Θ ( g ( n )) ( f of n is big-Theta of g of n ), if and only if, there exist positive real numbers A, B and kr such that

1. f is of order at least g, written f (n) is Ω ( g (n)) ( f of n is big-Omega of g of n ), if and only if, there exist positive real numbers A and ar such that

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

2. f is of order at most g, written f (n) is O ( g ( n )) ( f of n is big- O of g of n ), if and only if, there exist positive real numbers B and br such that
3. 0f(n)Bg(n) for every integer nb .

Proof:

Firstly, to show that 5n3+65n+30 is Ω(n3), we need to show that 5n3+65n+30 is greater than or equal to a positive multiple of n3 for all values of n that are sufficiently large. Now

5n35n3+65n+30 for every integer n1

Because when n1, 65n+30 is positive. Thus we can let A=5 and a=1 to obtain

An35n3+65n+30 for every integer na ,

Concluding by definition of Ω -notation, that 5n3+65n+30 is Ω(n3).

Secondly, to show that 5n3+65n+30 is O(n3), we need to show that 5n3+65n+30 is greater than or equal to 0 and less than or equal to some positive multiple of n3 for all values of n that are sufficiently large. First note that because all terms of 5n3+65n+30 are positive,

05n3+65n+30 for every integer n1

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