• Given f(n) E 0(n), prove that f(n) E O(n²). • Given f(n) E O(n) and g(n) E O(n²), prove that f(n)g(n) E O(n³).
• Given f(n) E 0(n), prove that f(n) E O(n²). • Given f(n) E O(n) and g(n) E O(n²), prove that f(n)g(n) E O(n³).
Computer Networking: A Top-Down Approach (7th Edition)
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Author:James Kurose, Keith Ross
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Chapter1: Computer Networks And The Internet
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Problem R1RQ: What is the difference between a host and an end system? List several different types of end...
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![Below is a transcription of the image for an educational website. The content includes statements regarding Big O and Big Theta notation, often used in the analysis of algorithms.
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**Big O and Big Theta Notation: Proof Examples**
1. **Prove that \( f(n) \in \Theta(n) \) implies \( f(n) \in O(n^2) \).**
To demonstrate this, consider the definitions of \(\Theta(n)\) and \(O(n^2)\):
- By definition, \( f(n) \in \Theta(n) \) means there exist positive constants \(c_1\), \(c_2\), and \(n_0\) such that for all \(n \geq n_0\),
\[
c_1 \cdot n \leq f(n) \leq c_2 \cdot n
\]
- We need to show that \( f(n) \in O(n^2) \), which means there exist constants \(c\) and \(n_1\) such that for all \(n \geq n_1\),
\[
f(n) \leq c \cdot n^2.
\]
Since \(c_2 \cdot n \leq c_2 \cdot n^2\) for \(n \geq 1\), we can choose \(c = c_2\) and \(n_1 = \max(n_0, 1)\). Hence, \( f(n) \in O(n^2) \).
2. **Prove that if \( f(n) \in O(n) \) and \( g(n) \in O(n^2) \), then \( f(n)g(n) \in O(n^3) \).**
Here’s the breakdown:
- By definition, \( f(n) \in O(n) \) means there exist constants \(c_f\) and \(n_0\) such that for all \(n \geq n_0\),
\[
f(n) \leq c_f \cdot n.
\]
- Similarly, \( g(n) \in O(n^2) \) means there exist constants \(c_g\) and \(n_1\) such that for all \(n \ge](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8b914819-4152-4353-8ebc-b14bc0f05ba7%2F661e6185-c7ee-4846-909e-a1cc0fd6f7f6%2Fr58ss_processed.png&w=3840&q=75)
Transcribed Image Text:Below is a transcription of the image for an educational website. The content includes statements regarding Big O and Big Theta notation, often used in the analysis of algorithms.
---
**Big O and Big Theta Notation: Proof Examples**
1. **Prove that \( f(n) \in \Theta(n) \) implies \( f(n) \in O(n^2) \).**
To demonstrate this, consider the definitions of \(\Theta(n)\) and \(O(n^2)\):
- By definition, \( f(n) \in \Theta(n) \) means there exist positive constants \(c_1\), \(c_2\), and \(n_0\) such that for all \(n \geq n_0\),
\[
c_1 \cdot n \leq f(n) \leq c_2 \cdot n
\]
- We need to show that \( f(n) \in O(n^2) \), which means there exist constants \(c\) and \(n_1\) such that for all \(n \geq n_1\),
\[
f(n) \leq c \cdot n^2.
\]
Since \(c_2 \cdot n \leq c_2 \cdot n^2\) for \(n \geq 1\), we can choose \(c = c_2\) and \(n_1 = \max(n_0, 1)\). Hence, \( f(n) \in O(n^2) \).
2. **Prove that if \( f(n) \in O(n) \) and \( g(n) \in O(n^2) \), then \( f(n)g(n) \in O(n^3) \).**
Here’s the breakdown:
- By definition, \( f(n) \in O(n) \) means there exist constants \(c_f\) and \(n_0\) such that for all \(n \geq n_0\),
\[
f(n) \leq c_f \cdot n.
\]
- Similarly, \( g(n) \in O(n^2) \) means there exist constants \(c_g\) and \(n_1\) such that for all \(n \ge
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