Question

(a) Show that the set of all finite sequences of elements of the one-element set \\{0\\} is countably infinite. (b) Show that the set of all finite sequences of elements of the two-element set (0,1] is countably infinite. (c) Challenge: Show that the set of all finite sequences of natural numbers is countably infinite. (Hint: Use a diagonal argument.)

Short answer

All sets of finite sequences described are countably infinite by mapping them to the natural numbers.

Step-by-step solution

Understand the problem and definitions

The problem asks us to show that certain sets are countably infinite. Countably infinite means there is a one-to-one correspondence between the set and the natural numbers, \(\mathbb{N}\). A finite sequence means a sequence with a limited number of elements.

Solve part (a)

For the set \( \{0\} \), every sequence of finite length is made only of 0's. Let the length of a sequence be \( n \), where \( n \) is a natural number. There is exactly one sequence for each natural number \( n \), e.g., for \( n = 1 \), the sequence is \( \{0\} \), for \( n = 2 \), the sequence is \( \{0,0\} \), and so on. Since there is a sequence for every natural number and vice versa, this set is countably infinite.

Solve part (b)

For the set \((0,1]\), each element of a sequence can be either 0 or 1. For a sequence of length \( n \), there are \( 2^n \) possible sequences (each position has two choices: 0 or 1). For any positive integer \( n \), there are \( 2^n \) sequences. Pair each sequence with a unique natural number, e.g., start with sequences of length 1, then length 2, and so on. This shows the sequences can be mapped to \( \mathbb{N} \), making this set countably infinite.

Solve part (c)

The set of all finite sequences of natural numbers \( \mathbb{N} \) involves sequences where each position can hold any natural number. We can use a diagonal argument here: consider an infinite matrix where row \( i \) represents sequences with length \( i \). Row 1 includes (1),(2),...; row 2 includes (1,1), (1,2), (2,1), ...; and so forth. Enumerate sequences by diagonals. By covering all possible sequences in this way, each finite sequence corresponds to a natural number, proving that this set is countably infinite.

Key concepts

Finite Sequences

Finite sequences are ordered lists that end. You can think of them as a list with a start and an end, having a specific number of elements. For example, in a sequence with the one-element set \( \{0\} \), you could have sequences like (0), (0,0), and (0,0,0). Each of these sequences only has zero in them, no matter how long they are.

• In finite sequences, the length is limited by some natural number \( n \).
• Sequences build automatically by incrementing the length.
• This step-by-step growth continues indefinitely.

The idea of finite sequences extends to more complex sets, like the two-element set \( (0, 1] \). Here, each element in a sequence can be 0 or 1, opening up more possibilities.
For a sequence length \( n \), there are \( 2^n \) possible sequences, demonstrating why this collection is considered infinite. Each sequence has a specific order that is important, and it will not change once set.

Diagonal Argument

The diagonal argument is a clever way to show something about infinite sets. This technique helps us make sense of how infinite sequences relate to the set of natural numbers.
You'll often see the diagonal argument used to prove that certain infinite sets are countable. Here's how it works in the context of finite sequences of natural numbers:

• Think of an infinite grid or table where each element represents part of a sequence.
• Each sequence is represented in a row of the table, as you lengthen the sequences you're essentially making the grid bigger.
• By listing these sequences diagonally across the table, you cover every possible combination.

This diagonal enumeration ensures every finite sequence matches up with at least one natural number, proving countability. This is a bit technical, but imagine covering all combinations in a single, never-ending sequence—this assures that no sequence is left out.

Natural Numbers

Natural numbers are the building blocks of many mathematical ideas. They are the numbers you count with: 1, 2, 3, and so on. These numbers are infinite—there is no largest natural number. Here’s why they’re crucial in understanding countably infinite sets:

• Natural numbers can be paired uniquely with many types of sets, showing both finite and infinite expansions.
• They are simple to list out, one-by-one, making them perfect for creating mappings with elements of other sets.

When determining if a set is countably infinite, we try to list it in a way that is one-to-one with natural numbers. If we manage to do that, just as with the set of finite sequences, this means they line up perfectly with \( \mathbb{N} \). This so-called 'perfect line-up' highlights their utility in showing how some infinite sets aren't bigger than natural numbers, even if they seem like they might be.