Proving the Equivalence Between ( fn O(gn) ) and ( gn Omega(fn) ) for Functions with Domain ( {1, 2, 3, ldots} )

When dealing with asymptotic analysis, particularly with the Big O and Big Omega notations, it is often necessary to establish the relationship between these notations for given functions. Specifically, we may need to prove that ( fn O(gn) ) implies ( gn Omega(fn) ). This article will demonstrate the proof of this equivalence with clear definitions and logical steps, ensuring a thorough understanding of the concept.

Definitions of Big O and Big Omega Notations

To begin, let's recall the formal definitions of Big O and Big Omega notations:

Big O Notation

We say ( fn O(gn) ) if there exist positive constants ( C ) and ( n_0 ) such that for all ( n geq n_0 ), the following holds:

fn leq C cdot gn

Big Omega Notation

We say ( gn Omega(fn) ) if there exist positive constants ( c ) and ( n_1 ) such that for all ( n geq n_1 ), the following holds:

gn geq c cdot fn

Proof of the Equivalence

Given the assumption ( fn O(gn) ), we start by expressing this as:

fn leq C cdot gn quad text{for all } n geq n_0

We aim to show that ( gn Omega(fn) ). This means we need to find constants ( c ) and ( n_1 ) such that:

gn geq c cdot fn quad text{for all } n geq n_1

Rearranging the Inequality

From the definition of ( fn O(gn) ), we can rearrange the inequality:

gn geq frac{1}{C} cdot fn

Lets choose ( c frac{1}{C} ). Therefore, we have:

gn geq c cdot fn quad text{for all } n geq n_0

Choosing ( n_1 )

To establish this inequality, we can choose ( n_1 n_0 ). This means that for all ( n geq n_1 ), the following holds:

gn geq c cdot fn

Conclusion

Since we have shown that there exist positive constants ( c ) and ( n_1 ) such that:

gn geq c cdot fn quad text{for all } n geq n_1

we can conclude that:

gn Omega(fn)

Thus, we have proved that if ( fn O(gn) ), then ( gn Omega(fn) ).

This proof demonstrates the fundamental relationship between Big O and Big Omega notations and is crucial in understanding the behavior of functions as ( n ) approaches infinity in algorithmic analysis and theoretical computer science.