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Chapter 2: Problem 4

Show that a) the set of increasing sequences of natural numbers \(\left\\{n_{1}

Short Answer

Expert verified
Question: Prove that a) the set of increasing sequences of natural numbers has the same cardinality as the set of decimal fractions between 0 and 1, and b) the set of all subsets of a countable set has the cardinality of the continuum. Answer: a) We can define a bijection between increasing sequences of natural numbers and decimal fractions between 0 and 1 by mapping each increasing sequence to a decimal fraction using the function: \(f\left(n_1, n_2, n_3, \dots\right) = 0.n_1n_2n_3\dots\). This function is both injective and surjective, so it represents a bijection between the two sets. b) Using Cantor's Diagonal Argument, we can prove that the set of all subsets of a countable set is uncountable. Since we know from part a) that the set of increasing sequences of natural numbers has the same cardinality as the set of decimal fractions between 0 and 1 (the cardinality of the continuum), we can deduce that the set of all subsets of a countable set also has the cardinality of the continuum.

Step by step solution

01

Find a function between increasing sequences and decimal fractions#

Let's define a function \(f:\mathbb{N}^\ast\to (0,1)\) as follows:\\ \(f\left(n_1, n_2, n_3, \dots\right) = 0.n_1n_2n_3\dots\)\\ where \(\mathbb{N}^\ast\) are the increasing sequences of natural numbers, and \(n_i\) are the natural numbers in the sequence. Now let's prove this function is a bijection:
02

Prove the function is injective (one-to-one)#

Assume we have two different sequences, say: \((n_1, n_2, n_3, \dots)\) and \((m_1, m_2, m_3, \dots)\), with at least one element different. Since the two sequences are different, there exists a smallest index \(k\) such that \(n_k \neq m_k\). Since they are different in the k-th position, \(f(n_1, n_2, \dots) \neq f(m_1, m_2, \dots)\). Therefore, the function \(f\) is injective.
03

Prove the function is surjective (onto)#

Let \(x=0.x_1x_2x_3 \dots\) be a decimal fraction between 0 and 1. We want to find an increasing sequence of natural numbers \((n_1, n_2, n_3, \dots)\) such that \(f(n_1, n_2, n_3, \dots) = x\). The increasing sequence we are looking for is simply \((x_1, x_2, x_3, \dots)\). Therefore, the function \(f\) is surjective. Since the function \(f\) is both injective and surjective, it is a bijection. Hence, the set of increasing sequences of natural numbers has the same cardinality as the set of decimal fractions between 0 and 1.
04

Prove the set of all subsets of a countable set is uncountable (Cantor's Diagonal Argument)#

Let \(A\) be a countable set, and let \(P(A)\) be the set of all subsets of \(A\). Assume there is a bijection \(g: A \to P(A)\). Now define a set \(B \subseteq A\) such that \(x \in B\) if \(x \notin g(x)\). This set \(B\) cannot be equal to any \(g(x)\) since \(x \in B\) iff \(x \notin g(x)\) for all \(x \in A\). This contradicts our assumption that there is a bijection between \(A\) and \(P(A)\). Hence, there cannot be a bijection between \(A\) and \(P(A)\). Therefore, the set of all subsets of a countable set is uncountable.
05

Show the set of all subsets of a countable set has cardinality of the continuum#

Since we have already proven that the set of increasing sequences of natural numbers has the same cardinality as the set of decimal fractions between 0 and 1, the cardinality of the set of increasing sequences is equal to the cardinality of the continuum. Now let's take a countable set \(A\). The set of all finite increasing sequences of elements from the set \(A\) is also countable. The set of all infinite increasing sequences can be put in bijection with the set of increasing sequences of natural numbers. Hence, the cardinality of the set of all subsets of a countable set is equal to the cardinality of the continuum.

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Key Concepts

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

Cardinality
In mathematical analysis, the concept of _cardinality_ explores the size of sets. It doesn't always refer to the number of elements in a set as you might expect. Instead, cardinality uses the idea of bijections to compare sizes. Two sets have the same cardinality if there exists a bijection between them. For instance, both the set of natural numbers, \( \mathbb{N} \), and the set of fractions between 0 and 1 share the same cardinality. This is partly because you can pair elements in both sets without any leftover elements. This form of pairing uses a bijection, confirming the sets' equal "sizes," despite their infinite nature.
Bijection
A _bijection_ is a type of function central to the concept of cardinality. It is a relationship between elements of two sets that are one-to-one and onto:
  • _One-to-One (Injective)_: Each element of the first set maps to a unique element of the second set.
  • _Onto (Surjective)_: Every element in the second set is mapped to by an element in the first set.
Bijection ensures a perfect pairing between two sets' elements, proving they have the same cardinality. In the context of increasing sequences and decimal fractions, we mapped sequences like \((n_1, n_2, \dots)\) to fractions \(0.n_1n_2\dots\). Proving both injectivity and surjectivity verified our bijection, showing equal cardinality between these sets.
Cantor's Diagonal Argument
_Cantor's Diagonal Argument_ is a clever technique used to demonstrate that some infinities are larger than others. This argument is crucial in proving that the set of all subsets of a countable set (powerset) is uncountable.
It works by assuming a list of elements (like binary strings) represents all possible subsets. We then construct a new subset by changing the diagonal elements to ensure it wasn't in the original list. This contradiction shows that no complete list of subsets exists. Consequently, the powerset of any infinite set is strictly larger than the set itself, showing that the continuum (the number of real numbers) has a greater cardinality than just countable sets like integers or natural numbers.
Increasing Sequences
In mathematics, an _increasing sequence_ is a sequence of numbers where each number is larger than the one before. More formally, a sequence \((n_1, n_2, n_3, \ldots)\) is increasing if \(n_1 < n_2 < n_3 < \ldots\).
Such sequences of natural numbers can be mapped to decimal fractions between 0 and 1. For example, the increasing sequence \((1, 2, 3, \dots)\) could correspond to the fraction 0.123... This correspondence is essential in proving that the set of all increasing sequences of natural numbers shares the same cardinality as the set of fractions like \(0.\alpha_1\alpha_2\dots\). These sequences are crucial for connecting discrete mathematics with continuous fractions.
Continuum
The term _continuum_ in mathematical analysis often refers to the "size" or cardinality of the real numbers. The continuum is denoted by \(2^{\aleph_0}\), representing the cardinality of all real numbers in an interval, say between 0 and 1.
The real power of this idea is evident when comparing the cardinality of real numbers to countable sets. The continuum is *larger* than any countable set, like the set of natural numbers or even the rational numbers. Through bijection and constructs like Cantor's Diagonal Argument, we see the stark differences in "infinite sizes," showcasing the beauty and complexity of mathematical infinity. This forms a deeper understanding of how mathematical sets and sequences relate to one another, highlighting the expansive possibilities of mathematical analysis.

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Most popular questions from this chapter

a) Show that along with the unique representation of an integer as $$ \left(\alpha_{n} \alpha_{n-1} \ldots \alpha_{0}\right)_{3} $$ where \(\alpha_{i} \in\\{0,1,2\\}\), it can also be written as $$ \left(\beta_{n} \beta_{n-1} \ldots \beta_{0}\right)_{3} $$ where \(\beta \in\\{-1,0,1\\}\). b) What is the largest number of coins from which one can detect a counterfeit in three weighings with a pan balance, if it is known in advance only that the counterfeit coin differs in weight from the other coins?

Show that a) from a system of closed intervals covering a closed interval it is not always possible to choose a finite subsystem covering the interval; b) from a system of open intervals covering an open interval it is not always possible to choose a finite subsystem covering the interval; c) from a system of closed intervals covering an open interval it is not always possible to choose a finite subsystem covering the interval.

Verify that the rational numbers \(\mathbb{Q}\) satisfy all the axioms for real numbers except the axiom of completeness.

a) Verify that \(\mathbb{Z}\) and \(\mathbb{Q}\) are inductive sets. b) Give examples of inductive sets different from \(\mathbb{N}, \mathbb{Z}, \mathbb{Q}\), and \(\mathbb{R}\).

Show that a) every infinite set contains a countable subset; b) the set of even integers has the same cardinality as the set of all natural numbers; c) the union of an infinite set and an at most countable set has the same cardinality as the original infinite set; d) the set of irrational numbers has the cardinality of the continuum; e) the set of transcendental numbers has the cardinality of the continuum.
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