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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 for \(n \in \mathbb{N}\) and \(a>0\) the equation \(x^{n}=a\) has a positive root (denoted \(\sqrt[n]{a}\) or \(a^{1 / n}\) ). b) Verify that for \(a>0, b>0\), and \(n, m \in \mathbb{N}\) $$ \sqrt[n]{a b}=\sqrt[n]{a} \cdot \sqrt[n]{b} \quad \text { and } \quad \sqrt[n]{\sqrt[m]{a}}=\sqrt[n \cdot m]{a} $$ c) \(\left(a^{\frac{1}{n}}\right)^{m}=\left(a^{m}\right)^{\frac{1}{n}}=: a^{m / n}\) and \(a^{1 / n} \cdot a^{1 / m}=a^{1 / n+1 / m}\). d) \(\left(a^{m / n}\right)^{-1}=\left(a^{-1}\right)^{m / n}=: a^{-m / n}\). e) Show that for all \(r_{1}, r_{2} \in \mathbb{Q}\) $$ a^{r_{1}} \cdot a^{r_{2}}=a^{r_{1}+r_{2}} \quad \text { and } \quad\left(a^{r_{1}}\right)^{r_{2}}=a^{r_{1} r_{2}} $$

a) Show that the inclusion relation is a partial ordering relation on sets (but not a linear ordering!). b) Let \(A, B\), and \(C\) be sets such that \(A \subset C, B \subset C, A \backslash B \neq \varnothing\), and \(B \backslash A \neq \varnothing\). We introduce a partial ordering into this triple of sets as in a). Exhibit the maximal and minimal elements of the set \(\\{A, B, C\\}\). (Pay attention to the non-uniqueness!)

a) How many different numbers can one define using 20 decimal digits (for example, two ranks with 10 possible digits in each)? Answer the same question for the binary system. Which system does a comparison of the results favor in terms of efficiency? b) Evaluate the number of different numbers one can write, having at one's disposal \(n\) digits of a \(q\)-ary system. (Answer: \(q^{n / q} .\) ) c) Draw the graph of the function \(f(x)=x^{n / x}\) over the set of natural- number values of the argument and compare the efficiency of the different systems of computation.

Using the principle of induction, show that a) the sum \(x_{1}+\cdots+x_{n}\) of real numbers is defined independently of the insertion of parentheses to specify the order of addition; b) the same is true of the product \(x_{1} \cdots x_{n}\) c) \(\left|x_{1}+\cdots+x_{n}\right| \leq\left|x_{1}\right|+\cdots+\left|x_{n}\right|\) d) \(\left|x_{1} \cdots x_{n}\right|=\left|x_{1}\right| \cdots\left|x_{n}\right|\) e) \(((m, n \in \mathbb{N}) \wedge(m-1\) and \(n \in \mathbb{N}\), equality holding only when \(n=1\) or \(x=0\) (Bernoulli's inequality); g) \((a+b)^{n}=a^{n}+\frac{n}{1 !} a^{n-1} b+\frac{n(n-1)}{2 !} a^{n-2} b^{2}+\cdots+\frac{n(n-1) \cdots 2}{(n-1) !} a b^{n-1}+b^{n}(\) Newton's binomial formula).

What is the smallest number of questions to be answered "yes" or "no" that one must pose in order to be sure of determining a 7 -digit telephone number?
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