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Chapter 3: Problem 48

Prove each. The set of irrational numbers is uncountable. (Hint: Prove by contradiction.)

Short Answer

Expert verified
To prove that the set of irrational numbers is uncountable, we assume it is countable and that there is a one-to-one correspondence with natural numbers. We list the irrational numbers in a sequence and write their decimal representations. Using Cantor's diagonal argument, we define a new decimal number \(y\) by taking the diagonal elements of the matrix formed by the decimal representations of our listed irrational numbers and adding 1 to each of these (mod 10). Since \(y\) has different decimal representation than any irrational number in our list, it is either an irrational or a rational number. However, its non-terminating and non-repeating decimal representation denotes it as an irrational number. Therefore, there exists an irrational number not in our sequence, contradicting our original assumption. Thus, the set of irrational numbers is uncountable.

Step by step solution

01

Assume the set of irrational numbers is countable

Assume that the set of irrational numbers is countable. By definition, if a set is countable, there exists a bijective function (one-to-one correspondence) from the set of natural numbers (N) to the set of irrational numbers (let's denote it as I). In other words, every irrational number can be associated with a unique natural number, and vice versa.
02

List all irrational numbers in a sequence

We will demonstrate that there exists a new irrational number which cannot be in our list, using Cantor's diagonal argument. First, list the first few irrational numbers in a sequence using the assumed one-to-one correspondence with natural numbers. Let's call these irrational numbers \(x_1, x_2, x_3, \dots\).
03

Consider the decimal representation of each irrational number

Since irrational numbers cannot be expressed as a ratio of two integers, their decimal representations are non-terminating and non-repeating. Write the decimal representation of each irrational number in the sequence: \(x_1 = 0.a_{11}a_{12}a_{13}\dots\) \(x_2 = 0.a_{21}a_{22}a_{23}\dots\) \(x_3 = 0.a_{31}a_{32}a_{33}\dots\) \(\vdots\) Where each \(a_{ij}\) is a number between 0 and 9.
04

Define a new decimal using the diagonal elements of the matrix

We will now define a new decimal number by taking the diagonal elements of the matrix formed by the decimal representations of our listed irrational numbers. We will also add 1 to each of these diagonal elements (modulo 10 to keep the number between 0 and 9). Let's call this new decimal number \(y\): \(y = 0.(a_{11} + 1)(a_{22} + 1)(a_{33} + 1)\dots\) (mod 10)
05

Prove that the new decimal number is irrational

Since the new decimal number \(y\) has been constructed to have a different decimal representation than any irrational number in our original list (as every digit differs from the corresponding digit in the diagonal), it cannot belong to our list. Therefore, it is either an irrational or a rational number. However, its decimal representation is non-terminating (as its digits are formed from the non-terminating decimal representations of the irrational numbers) and non-repeating (as every digit differs from the diagonal digits). Therefore, \(y\) must be an irrational number.
06

Contradiction

We have now shown that there exists an irrational number (namely, \(y\)) not in our sequence of irrational numbers. However, this contradicts our original assumption that the set of irrational numbers is countable and has a one-to-one correspondence with the set of natural numbers. As the assumption led to a contradiction, we can conclude that the set of irrational numbers is uncountable.

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Key Concepts

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

Irrational Numbers
Irrational numbers are numbers that cannot be written as a simple fraction, or ratio of two integers. This is because their decimal expansion is non-terminating and non-repeating. Examples of irrational numbers include \( \sqrt{2} \), \( \pi \), and \( e \). These numbers have a unique property: they go on forever after the decimal point without repeating any pattern.

Unlike rational numbers, which have neat fractional representations, irrational numbers can't be pinned down in the same way.
  • This makes them an important part of mathematics, as they fill in the gaps between rational numbers on the number line.
  • They form the set known as the real numbers along with rational numbers.
  • The realization that there are more irrational numbers than rational ones leads to deeper insights about the nature of numbers and infinity.
Cantor's Diagonal Argument
Cantor's Diagonal Argument is a clever method used to show that some infinite sets are larger than others. In our context, this argument demonstrates that the set of irrational numbers is uncountable, meaning it is so large that its elements cannot be paired one-to-one with the natural numbers.

The argument works by creating a list of numbers and constructing a new number that is guaranteed to be different from every number on the list. Here's how it works:
  • Imagine a list of numbers written in decimal form, each extending infinitely.
  • Create a diagonal by taking the first digit of the first number, the second digit of the second number, and so on. This forms a new sequence.
  • Alter each digit of this diagonal sequence, for instance, by adding 1 to each digit (modulo 10), to ensure the new number differs from each number in the list.
  • This process guarantees the new number is not on the list, thus proving the original list was incomplete.
Cantor's ingenious construction proves that no matter how you list out the irrationals, there will always be some left unlisted, showing the uncountable nature of such numbers.
Countable vs. Uncountable Sets
In mathematics, understanding the difference between countable and uncountable sets provides insight into the nature of infinity. A set is countable if its elements can be matched one-to-one with the natural numbers. This means even though the set might be infinite, its elements can be listed in a sequence or ordered like natural numbers.

Examples of countable sets include:
  • The set of all integers
  • The set of all rational numbers
On the other hand, a set is uncountable if it cannot be matched one-to-one with natural numbers. Uncountable sets are more "full" or "dense" in comparison. The set of irrational numbers is such an example.
  • Uncountable means that no matter how you try, you cannot list all the elements in a way that will match each with a natural number.
  • This concept leads to the idea that not all infinities are equal. Some infinities, like that of the irrational numbers, are significantly larger.
  • Understanding countable vs. uncountable sets helps mathematicians grasp the vastness of different types of infinity and their applications across mathematics.

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Most popular questions from this chapter

Determine if the functions are bijective. If they are not bijective, explain why. \(g: \Sigma^{*} \rightarrow \Sigma^{*}\) defined by \(g(w)=a w a,\) where \(\Sigma=\\{a, b, c\\}.\)

Let \(M\) denote the set of \(2 \times 2\) matrices over \(\mathbf{w} .\) Let \(f : N \rightarrow M\) defined by \(f(n)=\left[\begin{array}{ll}{1} & {1} \\ {1} & {0}\end{array}\right]^{n} .\) Compute \(f(n)\) for each value of \(n.\) $$4$$

Evaluate each sum, where \(\delta_{i j}\) is defined as follows. $$ \delta_{i j}=\left\\{\begin{array}{ll}{1} & {\text { if } i=j} \\ {0} & {\text { otherwise }}\end{array}\right. $$ \(\left[\delta_{j} \text { is called Kronecker's delta, after the German mathematician Leopold }\right.\) Kronecker \((1823-1891) . ]\) $$ \sum_{i=1}^{3} \sum_{j=1}^{i}(j+3) $$

Just as \(\sum\) is used to denote sums, the product \(a_{k} a_{k+1} \ldots a_{m}\) is denoted by \(\prod_{i=k}^{m} \mathrm{a}_{i} .\) The product symbol \(\Pi\) is the Greek capital letter \(p i .\) For example, \(n !=\prod_{i=1}^{n} i .\) Evaluate each product. $$\prod_{i=1}^{3}(i+1)$$

Let \(A=\left[\begin{array}{ccc}{1} & {0} & {-1} \\ {0} & {2} & {3}\end{array}\right], B=\left[\begin{array}{ccc}{0} & {-2} & {5} \\ {0} & {0} & {1}\end{array}\right],\) and \(C=\left[\begin{array}{ccc}{-3} & {0} & {0} \\\ {0} & {1} & {2}\end{array}\right]\) . Find each. $$3 A+(-2) B$$
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