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$A$ = $\{X\subseteq \mathbb N : |X| = \aleph_0\land |\bar X| \lt\aleph_0 \} $

I need to find the cardinality of group A.

As I understood, I need to find a function that helps me determine the cardinality.

Any ideas?

Gil
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  • I am not sure about it but I think that $A$ is the set of finite subsets of $\mathbb N$ – Kroki May 12 '19 at 15:40
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    I think that this problem is equivalent to counting the number of finite subsets of $\mathbb{N}$, since any subset is clearly uniquely determined by it's compliment. Try counting subsets of order 0, 1, 2, 3... and recall that a countable union of countable sets is countable. – jpmacmanus May 12 '19 at 15:42
  • @Youem Cofinite. – Fabio Somenzi May 12 '19 at 15:42
  • Complement set of X** @PeterForeman – Gil May 12 '19 at 15:47

1 Answers1

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Let $B_n = \{X\subset \mathbb N: |X| = n \}$, it is clear that $$|A| = \left|\bigcup_{n\in \mathbb N}B_n \right|.$$

Since $\left|\displaystyle\bigcup_{n\in \mathbb N}B_n \right| \ge \left|B_1\right| = |\mathbb N| = \aleph_0$ and $\left|B_n\right|\le \left|\mathbb N^n\right| = \aleph_0$, then $|A| = \aleph_0$

Kroki
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