The set of subsets of size $k$ in $\mathbb{N}$ is countable

Question

Let $P_{k}(\mathbb{N}) = \{ A \subset \mathbb{N} \ | \ |A|=k \}$. I want to prove that $P_k$ is countable for each $k$.

So I showed that this was a set of countable subsets, but I am not sure how to construct a one to one function to naturals.

I was also wondering how to prove that the set of all finite subsets of $\mathbb{N}$ is countable to make the proof easier, so I constructed the function $g = r^n$ from $r = 1$ to infinity where $n \in f$ where $f$ is a function that maps the elements of the finite countable sets into $\mathbb{N}$

Answer 1

An idea: take any element $\;X\in P_k(\Bbb N)\;$ and write its elements in ascending order:

$$X=\{a_1,a_2,...,a_k\} \;,\;\;a_1<a_2<\ldots <a_k$$

Now, with this agreement on, show that

$$f: P_k(\Bbb N)\to\Bbb N\;,\;\;f\left(\{a_1,...,a_k)\right):=\sum_{m=1}^k a_{m-1} 10^{m-1}$$

is an injection, and thus $\;\left|P_k(\Bbb N)\right|\le\aleph_0\;$

If by "countable" you also mean "infinite", show that

$$\;\left|\{\;\{1,2,...,k\}\,,\,\, \{1,2,...,k-1,k+1\}\;,\;\;\{1,2,...,k-1,k+2\}\,,\ldots\right|=\aleph_0$$

Answer 2

The easiest way is to use Cantor Bernstein Schroder. there is clearly an injection from the naturals to the set of subsets of cardinality k. We then define an injection the other way where $p_1,p_2,...,p_k$ are $k$ distinct primes and given a subset $\{a_1,a_2,...,a_k\}$ we map it to the number $p_1^{a_1},p_2^{a_2},...,p_k^{a_k}$ by the fundamental theorem of arithmetic (the uniqueness part) this assures us it's a bijection. You can also, by making a minor adjustment use this method to show that there is an injection from the set of finite sets of naturals to the naturals

Answer 3

Let $\varphi:\Bbb P_k(\Bbb N)\to \Bbb N^k$ by $\varphi(A)=(x_1,\ldots,x_k)$ with $A=\{x_1,\ldots,x_k\}$ and $x_1<\ldots<x_k$.

$\varphi$ is injective, and $\Bbb N^k$ is countable. Hence $\Bbb P_k(\Bbb N)$ is countable.

Answer 4

Isn't $$ P_k(\mathbb N) = \bigcup_l P_k^l(\mathbb N) $$ a countable union of countable sets, where $$ P_k^l(\mathbb N) = \{ A \subseteq \{1,\dots,l\} | |A|=k\} $$