Prove that the set of all 2-element subsets of $Z^+$ is denumerable.

Question

I am trying to prove:

Prove that the set S that has for elements all subsets of $Z^+$ containing exacly 2 elements is denumerable.

This is what I have:

Let $S_n=\left\{ \left\{ n, m\right\}|m>n\right\} $, that is $S_1=\left\{ \left\{ 1, 2\right\},\left\{1,3\right\},\left\{1,4\right\},...\right\} $, $S_2=\left\{ \left\{ 2, 3\right\},\left\{2,4\right\},\left\{2,5\right\},...\right\} $ and so on. Then, $∀n≥1$ $S_n$ is denumerable for $f:S_n→Z^+$ and $f\left\{n,m\right\}=m-n$. Since $S=⋃^∞_{n=1}S_n$ and the union of a denumerable set is denumerable, then $S$ is also denumerable.

I think I have the general idea but that I may be missing something. Can I get some feedback on if I have made an error?

Answer 1

Your attempt looks correct to me.

Here is an easier strategy:

Note that there is an obvious injection

$$S \to \mathbb{N}^2$$ which sends $\{n,m\}$ to $(n,m)$ if $n \leq m$.

Since $\mathbb{N}^2$ is countable, we can conclude.

Answer 2

You are absolutely right.

Alternatively you can see them as a subset of $\mathbb {N\times N}$.