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Powers. For all positive integers k, we have (f(n) + g(n))* e O((S(n))*) + O((g(n))*). (11.8)

Powers. For all positive integers k, we have (f(n) + g(n))* e O((S(n))*) + O((g(n))*). (11.8)

Database System Concepts
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
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I need hep proving the power rule of the Big-O using the definition of Big-O involving the absolute value and inequalitites. Thanks

**Powers.** For all positive integers \( k \), we have \[ (f(n) + g(n))^k \in O((f(n))^k) + O((g(n))^k). \] (11.8) This expression provides an asymptotic upper bound for the function \( (f(n) + g(n))^k \) using Big O notation. It essentially states that the growth rate of the sum raised to the power \( k \) is at most the sum of the separate growth rates of each term raised to the same power. This is useful in analyzing the complexity of algorithms where different functions contribute separately to the total complexity.
Transcribed Image Text:**Powers.** For all positive integers \( k \), we have \[ (f(n) + g(n))^k \in O((f(n))^k) + O((g(n))^k). \] (11.8) This expression provides an asymptotic upper bound for the function \( (f(n) + g(n))^k \) using Big O notation. It essentially states that the growth rate of the sum raised to the power \( k \) is at most the sum of the separate growth rates of each term raised to the same power. This is useful in analyzing the complexity of algorithms where different functions contribute separately to the total complexity.
Expert Solution
Proving using Induction :

Claim :

For all positive integers k , the claim is that :

 P(k) : (f(n) + g(n))k ∈ O(f(n)k) + O(g(n)k)

Base Case :

  1. For k = 1 , the P(1) : (f(n) + g(n)) ∈ O(f(n)) + O(g(n))
  2. P(1) is true .
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