Question

(a) Show that the set of all infinite sequences of elements of the one element set \(\\{0\\}\) is finite. (b) Show that the set of all infinite sequences of elements of the two element set \(\\{0.1\\}\) has the same cardinality as \(\mathcal{P}(\mathbb{N})\). (c) Challenge: Show that the set of all infinite sequences of elements of N' has the same cardinality as \(\mathcal{P}(\mathbb{N})\).

Short answer

(a) Countable; one sequence. (b) Same cardinality as \(\mathcal{P}(\mathbb{N})\). (c) Same as \(\mathcal{P}(\mathbb{N})\).

Step-by-step solution

Analyze Part (a)

The set \( \{0\} \) contains only one element, which is 0. An infinite sequence of elements from this set always consists of only zeros, i.e., \((0, 0, 0, \ldots)\). This means that there is exactly one such infinite sequence.

Conclusion for Part (a)

Since there is only one possible infinite sequence of zeros, the set of all infinite sequences of elements of \(\{0\}\) is not finite, but it is countable since it contains just one element: \[ \{ (0, 0, 0, \ldots) \} \].

Analyze Part (b)

The set \( \{0, 1\} \) has two elements, allowing each position in an infinite sequence to be either 0 or 1. The number of such sequences is equivalent to the number of functions from \(\mathbb{N}\) to \{0, 1\}. This can be mapped to the number of subsets of \(\mathbb{N}\).

Conclusion for Part (b)

The set of all infinite sequences of elements from \(\{0, 1\}\) is equivalent to the power set \(\mathcal{P}(\mathbb{N})\). By Cantor's theorem, the power set \(\mathcal{P}(\mathbb{N})\) has greater cardinality than \(\mathbb{N}\) and is uncountably infinite.

Analyze Part (c)

The set \(\mathbb{N}\) includes all natural numbers. An infinite sequence of elements from \(\mathbb{N}\) is a sequence of natural numbers \((n_1, n_2, n_3, \ldots)\).

Construct Mapping for Part (c)

One can map any subset \(S\) of \(\mathbb{N}\) to a unique sequence consisting of elements of \(S\), where the \(n\)-th term of the sequence is 1 if \(n\in S\) and 0 if \(notin S\). This demonstrates a one-to-one correspondence between the infinite sequences and \(\mathcal{P}(\mathbb{N})\).

Conclusion for Part (c)

The set of all infinite sequences of elements of \(\mathbb{N}\) also has the same cardinality as \(\mathcal{P}(\mathbb{N})\), which is uncountably infinite.

Key concepts

Infinite Sequences

Infinite sequences are fascinating in the world of mathematics. When you think about sets and sequences, it's helpful to consider element placement as if arranging an infinite list. Consider an infinite sequence using the set \( \{0\} \). Each entry in this sequence will always be zero, resulting in the single Infinite sequence example: \((0, 0, 0, \ldots)\).
However, when you use the set \( \{0, 1\} \), you start with a whole new level of possibilities. Every spot in the sequence could be a 0 or a 1. This freedom means your sequence can take on infinitely many forms, which makes exploring these infinite sequences exciting and complex. This idea bridges nicely into power sets.

Power Set

A power set is the set of all possible subsets of a given set. For a set \( S \), the power set is denoted as \( \mathcal{P}(S) \). When the set is \( \mathbb{N} \), the set of natural numbers, the power set is quite intriguing.
Consider the set \( \{0, 1\} \). This context leads you to create functions mapping natural numbers to 0 or 1. This exact mapping demonstrates the same size (cardinality) as the power set of natural numbers \( \mathcal{P}(\mathbb{N}) \). By Cantor's theorem, we understand that the power set is uncountably infinite, larger than the set of natural numbers themselves. This is why sequences from \( \{0, 1\} \) relate strongly to power sets.

Countability

Countability discusses whether a set can be matched with natural numbers. If you can list a set's elements in a sequence similar to counting, the set is countable. A fascinating result is how this applies to infinite sequences.
With the set \( \{0\} \), we find only one sequence exists, making this countable. But as discussed with \( \{0, 1\} \) or even natural numbers \( \mathbb{N} \), things get interesting. The infinite sequences related to these larger sets align with the power set of natural numbers, which is uncountably infinite.
This difference showcases how some infinite sets can go far beyond the concept of listable or countable sequences. Understanding this helps appreciate the depth and breadth of infinite sets, especially when dealing with power sets or sequences made from two-element sets.