Question

Using the proof of the Corollary 1 to Theorem 1 as a model, find a real number \(c\) and an \(N_{0} \in \mathbb{N}\) such that \(7 n^{4} \in O\left(n^{4}\right)\) for all \(n \in \mathbb{N}\) with \(n \geq N_{0}\)

Short answer

Choose \(c = 7\) and \(N_0 = 1\).

Step-by-step solution

Define Big O notation

Recall the Big O notation definition: A function \(f(n)\) is \(O(g(n))\) if there exist positive constants \(c\) and \(N_0\) such that for all \(n \geq N_0\), \(0 \leq f(n) \leq c \cdot g(n)\). In this exercise, we want to show that \(7n^4\) is \(O(n^4)\).

Identify Functions

We have \(f(n) = 7n^4\) and \(g(n) = n^4\). We need to find \(c\) and \(N_0\) such that \(0 \leq 7n^4 \leq c \cdot n^4\) for all \(n \geq N_0\).

Calculate Constant c

Since \(7n^4 \leq c \cdot n^4\) is true for any \(c \geq 7\), we will choose \(c = 7\). Thus, \(7n^4 \leq 7 \cdot n^4\).

Choose Minimum N_0

Since the inequality \(7n^4 \leq 7 \cdot n^4\) holds for all \(n \geq 1\), we can choose the smallest natural number \(N_0 = 1\).

Verify that Conditions are Met

For \(c = 7\) and \(N_0 = 1\), the condition \(0 \leq 7n^4 \leq 7 \cdot n^4\) holds for all natural numbers \(n\) such that \(n \geq 1\). The conditions for Big O notation are satisfied.

Key concepts

Understanding Asymptotic Analysis

Asymptotic analysis is a critical tool in computer science that helps in understanding the performance and efficiency of algorithms. By looking at the behavior of algorithms as the input size grows, we can make educated predictions on their performance without getting bogged down by constant factors or lower order terms.
This concept is particularly vital in determining how an algorithm will scale. For example, when comparing two algorithms, asymptotic analysis allows us to focus purely on the terms that grow the fastest as the input becomes very large. Thus, it helps in revealing the algorithm's complexity in a more abstract and universal manner.
When performing asymptotic analysis, one common approach is to use Big O notation. This notation describes an upper bound on the time complexity of an algorithm. It omits constant factors and lower-order terms, providing a high-level understanding of the algorithm's efficiency. Thus, it simplifies comparing and categorizing different algorithms based on their growth rates.

Exploring Algorithm Complexity

Algorithm complexity is a scaffold on which we build our understanding of how an algorithm performs. It involves measuring the resources needed by an algorithm, most commonly time and space, as a function of the size of the input data.
For time complexity, we often use Big O notation to describe the worst-case scenario of an algorithm's execution time. This notation helps in focusing only on the significant factors that affect performance, such as how the execution time increases as the input size grows. For example, a time complexity of \(O(n^2)\) implies that if the input size doubles, the algorithm's execution time quadruples.
Space complexity, on the other hand, considers the amount of memory required by an algorithm as the input size increases. A detailed understanding of both time and space complexities helps in choosing the right algorithm for a specific context or constraint.

Demystifying Proof Techniques

Proof techniques provide the foundation for rigorously demonstrating that a given algorithm performs as expected under defined conditions. In computer science, these techniques are crucial in substantiating claims about algorithm correctness and performance.
One common proof technique in algorithm analysis is induction. Inductive proofs often help in establishing the correctness of algorithms that rely on recursive patterns or processes. It involves proving that if a statement holds for a single case, it also holds for all subsequent cases.
Another vital proof technique is contradiction. This involves assuming that a statement is false, then showing that this assumption leads to a logical contradiction. This often helps to prove properties about algorithms or bounds on complexity.
Additionally, constructing proofs using Big O notation requires demonstrating the existence of specific constants \(c\) and \(N_0\), as shown in the original exercise, thereby ensuring that a function fits within the defined bounds for all larger input sizes.