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Chapter 4: Problem 2

(a) Can a finite, nonempty set be inductive? Explain. (b) Is the empty set inductive? Explain.

Short Answer

Expert verified
(a) No, a finite, nonempty set cannot be inductive. This is because if there is a largest element N in the set, it would require N+1 to also be in the set, which contradicts the finiteness of the set. (b) No, the empty set is not inductive. Since it does not contain any elements, it does not satisfy the condition that the number 1 must be in the set for it to be considered inductive.

Step by step solution

01

Part (a) Definition of Inductive Set

An inductive set is a set that satisfies the following two conditions: 1. The number 1 is in the set. 2. If an element n is in the set, then n + 1 is also in the set. Now let's see if a finite, nonempty set can be inductive.
02

Part (a) Examination of a Finite, Nonempty Set

Suppose we have a finite, nonempty set. Since the set is finite, there will be some largest element in the set, let's call it N. In an inductive set, if an element n is in the set, then n + 1 is also in the set. Therefore, if the set is inductive, N + 1 should also be in the set. However, since the set is finite, there cannot be any element larger than N present in the set. This means that N + 1 cannot be in the set. Because of this contradiction, a finite, nonempty set cannot be inductive.
03

Part (b) Definition of Empty Set

The empty set, denoted as 鈭�, is a set with no elements. Now let's examine if the empty set can be inductive.
04

Part (b) Examination of the Empty Set

Recall the two conditions for a set to be inductive: 1. The number 1 is in the set. 2. If an element n is in the set, then n + 1 is also in the set. Since the empty set has no elements, it does not contain the number 1. Therefore, the empty set does not satisfy the first condition of being an inductive set. As a result, the empty set is not inductive.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Finite Sets
In mathematics, a finite set is a collection of elements that has a limited number of members. You can think of it like a basket that can only hold a certain number of apples. Once the basket is full, you can't add any more apples without taking some out first. A finite set works in a similar way.
  • **Definition**: A set with a countable number of elements, meaning you can write a list of all the elements and count them.
  • **Examples**: Consider the set \( \{2, 4, 6, 8, 10\} \). This is a finite set because it contains exactly five elements.
  • **Important Property**: A finite set can never be inductive. In an inductive set, if an element \( n \) is in the set, then \( n + 1 \) should also be in the set. But in a finite set, there exists a largest element, say \( N \). Thus, \( N + 1 \) can't be in the set, leading to a contradiction if the set were supposed to be inductive.
Understanding finite sets helps build a foundation for more complex concepts in set theory.
Empty Set
The empty set鈥攄enoted by the symbol \( \emptyset \) or simply \( \{\} \)鈥攊s a set that contains no elements at all. It's like an empty box with nothing inside it.
  • **Uniqueness**: The empty set is unique. There is only one empty set because any set with no elements is considered identical to any other set with no elements.
  • **Role in Inductive Sets**: When we talk about inductive sets, the empty set does not qualify as one. An inductive set requires that the number 1 is included and also that if \( n \) is in the set, \( n + 1 \) is as well. Since \( \emptyset \) has no elements, it does not fulfill the requirement of including the number 1.
  • **Significance**: Despite its simplicity, the concept of the empty set is crucial in set theory as it serves as the foundation for defining and understanding more complex sets.
Set Theory
Set theory is the branch of mathematical logic that studies sets, which are collections of objects. It serves as a fundamental part of mathematics and is the basis for various other branches in the field.
  • **Basic Principles**: In set theory, objects can be grouped into sets, making it easier to handle large collections of objects systematically.
  • **Applications**: Set theory provides a foundation for various other mathematical disciplines, such as algebra, topology, and even computer science. It helps us understand concepts like functions and relations through the use of sets.
  • **Subsets and Power Sets**: A subset is a set entirely contained within another set. The power set of a set is a set that contains all possible subsets, including the empty set and the set itself.
  • **Importance**: Understanding the properties of different kinds of sets, like finite and infinite sets, is crucial to mastering more advanced mathematical topics.
Set theory's significance extends beyond mathematics; it plays a role in logic, philosophy, and even fields like linguistics, demonstrating how fundamental the idea of collections and their interactions are to human thought.

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Most popular questions from this chapter

Most of the work done in constructing a proof by induction is usually in proving the inductive step. This was certainly the case in Proposition 4.2 . However, the basis step is an essential part of the proof. Without it, the proof is incomplete. To see this, let \(P(n)\) be $$ 1+2+\cdots+n=\frac{n^{2}+n+1}{2} $$ (a) Let \(k \in \mathbb{N}\). Complete the following proof that if \(P(k)\) is true, then \(P(k+1)\) is true. Let \(k \in \mathbb{N}\). Assume that \(P(k)\) is true. That is, assume that $$ 1+2+\cdots+k=\frac{k^{2}+k+1}{2} $$ The goal is to prove that \(P(k+1)\) is true. That is, we need to prove that $$ 1+2+\cdots+k+(k+1)=\frac{(k+1)^{2}+(k+1)+1}{2} $$ To do this, we add \((k+1)\) to both sides of equation (1). This gives $$ \begin{aligned} 1+2+\cdots+k+(k+1) &=\frac{k^{2}+k+1}{2}+(k+1) \\ &=\cdots \end{aligned} $$ (b) Is \(P(1)\) true? Is \(P(2)\) true? What about \(P(3)\) and \(P(4) ?\) Explain how this shows that the basis step is an essential part of a proof by induction.

For which natural numbers \(n\) do there exist nonnegative integers \(x\) and \(y\) such that \(n=4 x+5 y ?\) Justify your conclusion.

In calculus, it can be shown that $$ \begin{array}{l} \int \sin ^{2} x d x=\frac{x}{2}-\frac{1}{2} \sin x \cos x+c \quad \text { and } \\ \int \cos ^{2} x d x=\frac{x}{2}+\frac{1}{2} \sin x \cos x+c \end{array} $$ Using integration by parts, it is also possible to prove that for each natural number \(n,\) $$ \begin{aligned} \int \sin ^{n} x d x &=-\frac{1}{n} \sin ^{n-1} x \cos x+\frac{n-1}{n} \int \sin ^{n-2} x d x \text { and } \\ \int \cos ^{n} x d x &=\frac{1}{n} \cos ^{n-1} x \sin x+\frac{n-1}{n} \int \cos ^{n-2} x d x \end{aligned} $$ (a) Determine the values of $$ \int_{0}^{\pi / 2} \sin ^{2} x d x $$ and $$ \int_{0}^{\pi / 2} \sin ^{4} x d x $$ (b) Use mathematical induction to prove that for each natural number \(n\), $$ \begin{aligned} \int_{0}^{\pi / 2} \sin ^{2 n} x d x &=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots(2 n)} \frac{\pi}{2} \text { and } \\ \int_{0}^{\pi / 2} \sin ^{2 n+1} x d x &=\frac{2 \cdot 4 \cdot 6 \cdots(2 n)}{1 \cdot 3 \cdot 5 \cdots(2 n+1)} \end{aligned} $$ These are known as the Wallis sine formulas. (c) Use mathematical induction to prove that $$ \begin{aligned} \int_{0}^{\pi / 2} \cos ^{2 n} x d x &=\frac{1 \cdot 3 \cdot 5 \cdots(2 n-1)}{2 \cdot 4 \cdot 6 \cdots(2 n)} \frac{\pi}{2} \quad \text { and } \\ \int_{0}^{\pi / 2} \cos ^{2 n+1} x d x &=\frac{2 \cdot 4 \cdot 6 \cdots(2 n)}{1 \cdot 3 \cdot 5 \cdots(2 n+1)} \end{aligned} $$ These are known as the Wallis cosine formulas.

Let \(y=\ln x\) (a) Determine \(\frac{d y}{d x}, \frac{d^{2} y}{d x^{2}}, \frac{d^{3} y}{d x^{3}},\) and \(\frac{d^{4} y}{d x^{4}}\). (b) Let \(n\) be a natural number. Formulate a conjecture for a formula for \(\frac{d^{n} y}{d x^{n}} .\) Then use mathematical induction to prove your conjecture.

For which natural numbers \(n\) is \(n^{2}<2^{n} ?\) Justify your conclusion.
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