Question

Describe a partition of \(\mathbb{N}\) that divides \(\mathbb{N}\) into \(\aleph_{0}\) countably infinite subsets.

Short answer

One possible partition of \(\mathbb{N}\) into \(\aleph_{0}\) countably infinite subsets is to distribute each natural number to sets based on their least significant digit.

Step-by-step solution

Understanding Countable Infiniteness

Review the concept of countable infiniteness. Essentially, a set is countably infinite if their elements can be put into a one-to-one correspondence with the set of all natural numbers.

Generating Partitions

Consider a partition as follows: Start by distributing the natural numbers to various sets using a distinct pattern. For instance, let's set each element \(n\) in the set \(S_m\) where \(m\) is the least significant digit of \(n\). This means the number 21,43,61 would go to set \(S_1\), and the number 104,52 would go to set \(S_4\), and so forth.

Check Countable Infiniteness

Consider each partition \(S_m\) and verify that they are countably infinite. Each \(S_m\) has numbers 0m, 1m, 2m, 3m and so on which are countably infinite since there is a one-to-one correspondence between these numbers and the set of natural numbers. There are 10 such sets (0 to 9) giving a total of \(\aleph_{0}\) subsets.

Key concepts

Partition of Natural Numbers

Imagine taking an enormous pile of numbered blocks, where each block represents a natural number. A partition of natural numbers is like organizing these blocks into different boxes based on a specific rule, such that every block belongs to one and only one box, and no box is left empty.

A \textbf{partition} is therefore a way to divide or 'split up' a set into disjoint subsets. In the context of the exercise, we look at the set of all natural numbers, denoted by \(\mathbb{N}\), and we're tasked with grouping \(\mathbb{N}\) into subsets wherein each subset is countably infinite. A countably infinite set is a set with the same cardinality (size) as \(\mathbb{N}\), meaning that its elements can be listed in a sequence with the first element, the second element, and so on.

For example, one such partition might group the natural numbers based on their least significant digit. By doing so, we create ten distinct subsets, such as \(S_0\) for numbers ending in 0, \(S_1\) for numbers ending in 1, and so on up to \(S_9\). Each of these subsets is infinitely large since there are an infinite number of natural numbers ending in any given digit, and yet each is countable because we can list them in a sequence.

One-to-One Correspondence

Now let's focus on the concept of one-to-one correspondence, which is a fundamental idea in set theory. This concept is a way to compare the sizes of two sets.

In a one-to-one correspondence, also known as a bijection, each element from one set is paired with exactly one element from another set, and vice versa. There are no unpaired elements in either set. It is as if you were setting up pairs for a dance, ensuring everyone has a partner, with no one left out.

Using the same block and box analogy, you would be creating pairs where each block from box \(S_m\) has a dancing partner from the blocks lined up as \(1, 2, 3, ...\) and so on. This notion is crucial when determining that the subsets \(S_m\) are countably infinite: for each element in \(S_m\), you can find a unique natural number, and for every natural number, there's a unique element in \(S_m\). That's the one-to-one correspondence, showing that the cardinality of each \(S_m\) is the same as that of \(\mathbb{N}\).

Set Theory

Finally, let's explore set theory, the mathematical science of sets, which are collections of objects. In set theory, we talk about all sorts of sets, whether finite or infinite, and the relationships between them.

Set theory provides a foundation for essentially all of modern mathematics. It deals with notions of union, intersection, difference, and complement of sets. It also considers functions between sets and operations like Cartesian products. Countable infiniteness and partitions are examples of concepts in set theory that help us understand the structure and size of sets.

Returning to our exercise, when we partition \(\mathbb{N}\), we are using the principles of set theory to understand how the infinite set of natural numbers can be organized into smaller, yet still infinite subsets \(S_m\) that reflect its countably infinite nature. This representation gets at the heart of set theory: comprehending the infinite and finding patterns that help make sense of vast collections of elements.