The set of all finite subsets of $\mathbb{R}_+$ is countable.

Question

I'm struggling to figure out how to prove that the set of all finite subsets of $\mathbb{R}_+$ is countable. I thought that it wasn't but a TA told me it was and I need to prove why it's countable. I don't even know how to start this proof. If it helps, I solved this problem with $\mathbb{Z}_+$ by saying when writing down first several subsets of $\mathbb{Z}_+$, you can clearly see a pattern that can be enumerated.

It is definitely wrong. Even singleton subsets of $\Bbb R_+$ are uncountable. — WhatsUp, Jul 25 '20 at 19:24

score 4 · Answer 1 · answered Jul 25 '20 at 19:24

4

It is false. That set contains all the singletons from $R^+$ which is itself uncountable. So the set must be uncountable. Also, if $R^+$ is replaced by $Z^+$ it is still false as the power set of $Z^+$ is still uncountable !

answered Jul 25 '20 at 19:24

2

For $\Bbb Z^+$ it holds, as only the finite subsets are considered. – Berci Jul 25 '20 at 19:34

The set of all finite subsets of $\mathbb{R}_+$ is countable.

1 Answers1