Prove that the set of the subsets of N of size n is countable

Question

How can I prove that the set of subsets of N of size n is countable? X∈P(N) | |X| = n. Where do I go on from here?

Also how can I prove that the set of all finite subsets of N is countable.

Answer 1

Consider the set of all finite binary sequences — all strings of $0$s and $1$s. There are a few ways to write this set; let's use $$ \{0,1\}^{<\omega} := \bigcup_{n<\omega} \{0,1\}^n. $$ Similarly, the set of all finite subsets of $\Bbb N$ is $$\mathcal{P}_{<\omega}(\Bbb N) := \bigcup_{n<\omega} \mathcal{P}_n(\Bbb N), $$ where $\mathcal{P}_n(X)$ is the set of subsets of $X$ of size $n$.

There is an injection $\{0,1\}^{<\omega} \to \Bbb N$. We can think of a finite binary sequence as a base 2 numeral, but possibly with $0$s "in front", "on the left", in the high-order positions: $b_n \dots b_1$. We can have $n=0$, the empty sequence $\varepsilon$. So multiple numerals correspond to the same integer ($\varepsilon_2 = 0_2 = 00_2, 10_2 = 010_2$, etc.) If we put a $1$ in front, however, we get an injection $$ b_n \dots b_1 \mapsto 2^{n} + \sum_{i=1}^n b_i 2^{i-1} $$

There is an injection $\mathcal{P}_{<\omega}(\Bbb N) \to \{0,1\}^{<\omega}$. Given a finite set $A$, let $$ h(A) = \max_{a\in A} (a + 1), $$ the least integer greater than everything in $A$. Thus $h(\emptyset) = 0$, and if $A\ne \emptyset$ then $h(A) = \max(A) + 1$. So $A$ can be represented uniquely by a finite indicator function defined on the integers less than $h(A)$ — in other words, a binary sequence of length $h(A)$, a member of $\{0,1\}^{h(A)}$: $$ A \mapsto (c_0 \dots c_{h(A) - 1}) \text{ where }c_i = \begin{cases} 1 &\text{if $i\in A$,} \\ 0 &\text{if $i\notin A$.} \\ \end{cases} $$ Aside: The image of this injection is the empty sequence together with all binary sequences that end in a $1$. This is a regular language, $\varepsilon + (\mathbf{0} + \mathbf{1})^* \mathbf{1}$.

Obviously, for every $n\ge 0$, there's an injection $\mathcal{P}_{n}(\Bbb N) \to \mathcal{P}_{<\omega}(\Bbb N)$, the inclusion map.

So all these sets are countable. In fact, except for $\mathcal{P}_{0}(\Bbb N)$, every set mentioned is countably infinite.