Discrete Mathematics With Applicat...

5th Edition

EPP + 1 other

ISBN: 9781337694193

Textbook Problem

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Prove Theorem 11.2.4: If f is a real-valued function defined on a set of nonnegative integers and f ( n ) is Ω ( n m ) . where m is a positive integer, then f ( n ) is not O ( n p ) for any positive real number p < m .

To determine

The proof of the given statement.

Explanation

Given information:

The given statement is,

“A real valued functions f is defined on the set of non-negative if f(n) is Ω(nm) where m is a positive integer, then f(n) is not f(n) is Ο(np) for any positive real number p<m ”.

Proof:

Let’s assume f(n)=

Ο(np) ,

Then, by the definition of Ο (Big oh notation) there exists two constants c1, n0 such that,

f(n)≤c1⋅np for some n≥n0 -----------

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