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Let: $$\{A \cup \mathbb{N}_\text{even} \mid A \subseteq \mathbb{N}_\text{odd} \}$$

Why does it's cardinality equal $\aleph$ ?

I tried to find a bijection to the interval $(0,1)$ but it didn't work out.

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Here's a bijection $$\begin{align}\mathcal P(\mathbb N)&\to\{\,A\cup \mathbb N_{\text{even}}\mid A\subseteq \mathbb N_{\text{odd}}\,\}\\S&\mapsto\{\,2s-1\mid s\in S\,\}\cup \mathbb N_{\text{even}}\end{align}$$ and recall that $|\mathcal P(\mathbb N)|=|\mathbb R|=\aleph$.

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  • $\begingroup$ Great. thank you sir. $\endgroup$ – AnnieOK Apr 26 '14 at 22:02
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HINT: Find a countably infinite set $X$ such that this set has a natural bijection with $\mathcal P(X)$.

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