I have $$ L = \{A \in \mathcal P(\mathbb N) | A \text{ co finite over } \mathbb N\}$$

How can I prove that $L$ is countable?


  • $\begingroup$ I've edited your post to format the math in it. Please verify that it says what you intended. $\endgroup$ – user61527 Apr 28 '14 at 5:35
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    $\begingroup$ Note there is a canonical bijection between the set of finite subsets, and the set of cofinite subsets. $\endgroup$ – Asaf Karagila Apr 28 '14 at 5:36

Hint: Find a bijection $$\{ A \in \mathcal{P}(\mathbb{N}) : A\ \text{cofinite} \} \longleftrightarrow \{ A \in \mathcal{P}(\mathbb{N}) : A\ \text{finite} \}$$ and use familiar facts about countable sets to deduce that the latter, hence the former, is countable.


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