find cardinality of A

Question

$A$ = $\{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?

Answer 1

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 $|A| = \aleph_0$