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

Mathematics Asked by simey on September 19, 2020

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.

One Answer

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 by The73SuperBug on September 19, 2020

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