Question

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

Short answer

The partition of \(\mathbb{N}\) into eight countably infinite subsets can be defined as \(S_0, S_1, S_2, S_3, S_4, S_5, S_6, S_7\), where \(S_i = \{n \in \mathbb{N} : n \mod 8 = i\}\) for \(0 \leq i \leq 7\).

Step-by-step solution

Define the Set

We start from the set of natural numbers \(\mathbb{N}\) which includes all positive integers.

Conceptualize the Partition

The task is to divide this set into eight countably infinite subsets. We can approach this by assigning each natural number n to a subset based on the value of \(n \mod 8\). This will ensure that each natural number is assigned to exactly one of the eight subsets.

Formulate the Subsets

Let's define eight subsets of \(\mathbb{N}\) labelled \(S_0, S_1, S_2, S_3, S_4, S_5, S_6, S_7\). Define \(S_i\) (for \(0 \leq i \leq 7\)) as follows: \(S_i = \{n \in \mathbb{N} : n \mod 8 = i\}\). So for example, \(S_0\) is the set of all natural numbers divisible by 8, \(S_1\) is the set of all natural numbers that leave a remainder of 1 when divided by 8, and so forth.

Verification

By this construction, every natural number belongs to exactly one of the subsets \(S_0, S_1, S_2, S_3, S_4, S_5, S_6, S_7\). These subsets form a partition of \(\mathbb{N}\). Moreover, each \(S_i\) is countably infinite since the operation \(n \mod 8 = i\) will be true for infinitely many natural numbers.

Key concepts

Countably Infinite Subsets

When learning about the magnificent world of numbers, one concept that comes up is the 'countably infinite subset.' A set that is countably infinite has the same size as the set of natural numbers, \(\mathbb{N}\). Imagine you're lining up all the elements of a set and matching each one with a unique natural number. If you can do this without leaving any element unpaired, the set is countably infinite. This means the set is infinite but not so vast that you can't count it, just like the natural numbers themselves.

But how does one actually construct such a set? Let's look at the exercise where we create eight distinct subsets of natural numbers that are each countably infinite. By using modular arithmetic, which assigns each number to a unique subset based on its remainder when divided by 8, we achieve this neat trick. Every one of these subsets, \(S_i\), is infinite because no matter how far you go in the sequence of natural numbers, you will always find more numbers that fit into each subset. They never run out, hence they're countably infinite.

Modular Arithmetic

Modular arithmetic, sometimes referred to as 'clock arithmetic,' is a system for numbers where after you reach a certain number, you wrap around back to the beginning. This concept is vital to understand the textbook exercise we're considering.

In our context, we're using the modulus of 8. So, think of a clock that only goes up to 8 instead of 12. When a natural number is divided by 8, the remainder is what we're interested in. Each natural number \(n\) will have a remainder (\(n \mod 8\)) that is between 0 and 7. We use these remainders to create our subsets within the set of natural numbers, each containing all numbers that have the same remainder when divided by 8.

Why is Modular Arithmetic Useful Here?

It's useful because it guarantees a neat, non-overlapping partition of the natural numbers. Every number will fit into one and only one of our eight different subsets, which helps to maintain the integrity of our partition.

Set Theory

Set theory is a fundamental language of mathematics that deals with collections of objects, which we call 'sets.' Think of it like a box where you can toss in anything: numbers, letters, or even other boxes. In our exercise, we focused on sets of numbers—specifically, the set of all natural numbers \(\mathbb{N}\).

The concept of partitioning a set, like in our exercise, is like organizing your music into different playlists. Each playlist (subset) has a different vibe (property), but every song (number) belongs exclusively to one playlist. This organization is crucial for establishing mathematical structures and functions in a clear, unambiguous way.

Understanding set theory enhances your ability to grasp how mathematicians categorize and comprehend the infinite realm of numbers. It's the backbone that supports the idea of countably infinite subsets and the application of modular arithmetic, helping us see that mathematics is not just about numbers—it's about patterns, structures, and the elegant language that describes them.