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Chapter 7: Problem 26

Prove that any infinite set contains a countably infinite subset.

Short Answer

Expert verified
Given an infinite set A, we construct a countably infinite subset B by creating a one-to-one correspondence between the elements of B and the natural numbers. Starting with B as an empty set, iteratively include an element from A that is not in B yet for each natural number. This process results in B being countably infinite, proving that any infinite set contains a countably infinite subset.

Step by step solution

01

Define the countably infinite subset

To start, let's define a countably infinite subset B of set A. We will construct B in such a way that there is a one-to-one correspondence between the elements of B and the natural numbers, denoted as \(N = \{1, 2, 3, \dots\}\).
02

Assume A is infinite

Since A is infinite, we know that for every finite subset C of A, there exists an element in A that is not in C. This will be useful in the construction of subset B.
03

Construct the countably infinite subset B of A

To construct the countably infinite subset B, we can use the following iterative process: a. Let B be an empty set at the beginning. b. Since A is infinite, there always exists an element in A that we haven't included in B. Select one such element, say a_1, and include it in B. c. Now repeat the above step for the rest of the natural numbers, i.e., for every natural number n > 1, select an element from A that is not in the subset B yet. Let this element be a_n and include it in B. By doing this for all natural numbers, we are creating a one-to-one correspondence between the natural numbers and the elements of B.
04

Prove that B is countably infinite

After constructing the subset B, we now need to show that it is indeed countably infinite. To do so, we need to establish a one-to-one correspondence between the set B and the natural numbers. Since we have constructed B using an element a_n for each natural number n, there is a one-to-one correspondence between B and the natural numbers. That is, no elements in B are repeated and each natural number has a corresponding element in B. Thus, we have shown that any infinite set A contains a countably infinite subset B.

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Key Concepts

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

Countably Infinite Subset
A countably infinite subset is a special type of subset that can be paired one-to-one with the natural numbers.

Imagine you have an infinite set A. From this set, we can always extract a subset B that is countably infinite.

This means that B behaves like a list where each element has a distinct position, just like numbers in a sequence. For example: 1, 2, 3, and so on.
  • The subset is never ending.
  • Each position in the list corresponds to a natural number.
  • There is no repetition within the subset.
The fascinating part is that no matter how enormous or endless set A is, we can find this kind of subset within it.
One-to-One Correspondence
One-to-one correspondence is like setting up a perfect match between two groups.

In the context of our exercise, it means matching the elements of our subset B with natural numbers.

This ensures that every element in subset B matches exactly one unique natural number, without any leftover or repeated pairings.
  • Each natural number has a unique partner in subset B.
  • No two natural numbers point to the same element in B.
  • This kind of pairing perfectly illustrates the concept of a countably infinite set.
A real-world analogy is like giving each person in a line a unique ticket number.
By doing so, you ensure clarity and order, free of confusion or duplicity.
Natural Numbers
Natural numbers are the basic building blocks in mathematics, starting from 1 and going on forever: 1, 2, 3, 4, and so on.
These numbers are like markers we use for counting.

In our discussion, they help us understand how a set can be infinite but still have countable elements.
  • The sequence of natural numbers is practical for pairing with elements in other infinite sets.
  • They represent the simplest, non-negative count we know.
  • The use of natural numbers assures us that the set has no gaps or breaks.
These numbers are key to defining and understanding countably infinite sets.
When we talk about matching things with natural numbers, we rely on their unending sequence and straightforward nature.

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Most popular questions from this chapter

Let \(S\) be the set of all strings of \(a\) 's and \(b^{\prime}\) 's. a. Define \(f: S \rightarrow \mathbf{Z}\) as follows: For each string \(s\) in \(f(s)=\left\\{\begin{array}{l}\text { the number of } b \text { 's to the left } \\ \text { of the left-most } a \text { in } s \\ 0 \quad \text { if } s \text { contains no } a \text { 's }\end{array}\right.\) Find \(f(a b a), f(b b a b)\), and \(f(b)\). What is the range of \(f\) ? b. Define \(g: S \rightarrow S\) as follows: For each string \(s\) in \(S\), $$ g(s)=\text { the string obtained by writing the } $$ characters of \(s\) in reverse order. Find \(g(a b a), g(b b a b)\), and \(g(b)\). What is the range of \(g\) ?

Define Floor: \(\mathbf{R} \rightarrow \mathbf{Z}\) by the formula Floor \((x)=\lfloor x\rfloor\), for all real numbers \(x\). a. Is Floor one-to-one? Prove or give a counterexample. b. Is Floor onto? Prove or give a counterexample.

The functions of each pair in \(9-11\) are inverse to each other. For each pair, check that both compositions give the identity function. \(F: \mathbf{R} \rightarrow \mathbf{R}\) and \(F^{-1}: \mathbf{R} \rightarrow \mathbf{R}\) are defined by \(F(x)=3 x+2, \quad\) for all \(x \in \mathbf{R}\) and \(F^{-1}(y)=\frac{y-2}{3}, \quad\) for all \(y \in \mathbf{R}\)

Find functions defined on the set of nonnegative integers that define the sequences whose first six terms are given below. a. \(1,-\frac{1}{3}, \frac{1}{5},-\frac{1}{7}, \frac{1}{9},-\frac{1}{11}\) b. \(0,-2,4,-6,8,-10\)

Observe that \(m o d\) and \(d i v\) can be defined as functions from \(\mathbf{Z}^{\text {nonseg }} \times \mathbf{Z}^{+}\)to \(\mathbf{Z}\). For each ordered pair \((n, d)\) consisting of a nonnegative integer \(n\) and a positive integer \(d\), let \(\bmod (n, d)=n\) mod \(d\) (the nonnegative remainder obtained when \(n\) is divided by \(d\) ). \(\operatorname{div}(n, d)=n \operatorname{div} d\) (the integer quotient obtained when \(n\) is divided by \(d\) ). Find each of the following: a. \(\bmod (67,10)\) and \(\operatorname{div}(67,10)\) b. \(\bmod (59,8)\) and \(\operatorname{div}(59,8)\) c. \(\bmod (30,5)\) and \(\operatorname{div}(30,5)\)
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