Partitioning an infinite set

Question

Can you partition an infinite set, into an infinite number of infinite sets?

Answer 1

Yes.

First, note that it's enough to do it for countably infinite sets. For if $X$ is uncountable and $Y$ is a countably infinite subset, then $Z = X-Y$ is also infinite. Then, if we divide $Y$ into infintely many infinite sets, we've divided $X$ into infinitely many infinite sets.

So, let's consider a countable set $Y$. Of course, we may as well label the elements of $Y$ as $0,1,2,...$. In short, I'm going to break $\mathbb{N}$ into infinitely many infinite sets.

First note that there are infinitely many prime numbers. Further, if $p$ and $q$ are prime numbers, and if $p^a = q^b$, then we must have $p=q$ and $a=b$. This follows from unique factorization of numbers.

So, for each prime $p$, consider the set $Y_p = \{ p^a$ with $a>0\in\mathbb{N}\}$. Then, e.g., $Y_2 = \{2,4,8,16,32,64,128,...\}$ and $Y_3 =\{3,9,27,81,243,...\}$.

Clearly, each $Y_p$ is infinite. Further, if $Y_p$ and $Y_q$ have anything in common, then by what we said before, we must have that $p=q$. In other words, for different primes $p$ and $q$, the sets $Y_p$ and $Y_q$ have no overlap.

Since there are infinitely many primes, there are infinitely many $Y_p$ with no overlap. Finally, let $Z$ be all the naturals which are NOT powers of a prime number. $Z$ is also infinite since, for example, it contains $2*3, 2*3^2, 2*3^3, 2*3^4,...$

Thus, we've dividied $\mathbb{N}$ into infinitely many infinite sets: $Z$ and each of the $Y_p$.

Answer 2

Certainly. Look at the Cantor pairing function that shows a correspondence between the naturals and pairs of naturals. You have $f:\mathbb{N}\mapsto \mathbb{N\times N}$ which is a bijection. All sets of the form $(x,y)$ for a given $x$ are infinite. So the sets of $n$ that go to $(x,y)$ for a given $x$ are infinite.

On second thought, this is too complicated. $(x,y) \in \mathbb{N \times N}$ is infinite. The sets of the form $(x,1)$ are infinite, as are $(x,2)$, as are $\ldots$. So we have divided an infinite set into an infinite number of infinite sets. The pairing function just projects this onto $\mathbb{N}$.

Answer 3

For another example with a somewhat different flavour, chop the real line up at integers, dividing it into countable union of unit intervals, each of which has uncountably many points:

$$ \mathbb{R} = \bigcup_{n \in \mathbb{Z}} [n,n+1)$$

Answer 4

A simple way to split the set of natural numbers.Take the sets $S_n$ of numbers with sum of digits $n$ for each all $n$. Obviously no sets overlap and each $S_n$ contain numbers of the form $1111..0000...$ where the string of 1's contain $n$ of them, and as many zeros as you wish

Answer 5

An (imo) beautiful practical example from Elementary Number Theory.

As you know, any equivalence relation, partitions a set.

Now, if you partition the Natural numbers $(\mathbb{N})$ = {1, 2, 3, ... } ( 0 of course NOT included ), based on the relation "Prime Signature", you have what you are looking for.

Based on the Main Theorem of Arithmetic we can map any natural number to two ordered lists:

$$ n = \Pi_{i=1}^{\omega(n)} p_i^{\alpha^{i}}. $$

For example: $12 = 2^2 3^1$, 30= $2^1 3^1 5^1$, and so on, where $12$ belongs to the partition $\{2,1\}$ and $30$ belongs to the partition $\{1,1,1\}$.

We call the ordered list of ${\alpha_{i}}'s$ the "Prime Signature". ( Example: $1000$ has signature $\{3,3\}$ ).

What makes this infinite partition so 'beautiful' is that every subset of $\mathbb{N}$ in this infinite partition corresponds to a partition of a natural number.