Prove or disprove each of the following statements. To prove a statement, you should provide a formal proof that is based on the definitions of the order notations. To disprove a statement, you can either provide a counter example and explain it or provide a formal proof. All functions are positive functions. Ex.) f(n) = (g(n)) ⇒ g(n) = O(f(n)) f(n) = (g(n)), for large values of n we have Mig(n) ≤ f(n) ≤ M₂g(n) for some M₁ and M₂. This means we have f(n) ≤ g(n) ≤ f(n), which shows g(n) = (f(n)). M₂ M₂ a) f(n) ‡ o(g(n)) and f(n) ‡ w(g(n)) ⇒ f(n) = O(g(n)) b) f(n) ≤ 0(h(n)) and h(n) ≤ 0(g(n)) ⇒ f(n) ≤ 0(1) € c) f(n) = (g(n)) ⇒ 2f (n) = (29(n))

Prove or disprove each of the following statements. To prove a statement, you should provide a formal proof that is based on the definitions of the order notations. To disprove a statement, you can either provide a counter example and explain it or provide a formal proof. All functions are positive functions. Ex.) f(n) = (g(n)) ⇒ g(n) = O(f(n)) f(n) = (g(n)), for large values of n we have Mig(n) ≤ f(n) ≤ M₂g(n) for some M₁ and M₂. This means we have f(n) ≤ g(n) ≤ f(n), which shows g(n) = (f(n)). M₂ M₂ a) f(n) ‡ o(g(n)) and f(n) ‡ w(g(n)) ⇒ f(n) = O(g(n)) b) f(n) ≤ 0(h(n)) and h(n) ≤ 0(g(n)) ⇒ f(n) ≤ 0(1) € c) f(n) = (g(n)) ⇒ 2f (n) = (29(n))

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Prove or disprove each of the following statements. To prove a statement, you should provide
a formal proof that is based on the definitions of the order notations. To disprove a statement,
you can either provide a counter example and explain it or provide a formal proof. All
functions are positive functions.
Ex.) f(n) (g(n)) ⇒ g(n) = O(f(n))
f(n) € (g(n)), for large values of n we have M₁g(n) ≤ f(n) ≤ M2g(n) for some M₁
and M₂. This means we have f(n) ≤ g(n) ≤ f(n), which shows g(n) = (f(n)).
a) f(n) o(g(n)) and f(n) & w(g(n)) ⇒ f(n) = O(g(n))
b) f(n) = (h(n)) and h(n) = (g(n)) ⇒
f(n) ΕΘ(1)
g(n)
c) f(n) (g(n)) ⇒ 2f(n) = (29(n))

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