Cardinality of a set. 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. 

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 $|\mathcal P(\mathbb N)|=|\mathbb R|=\aleph$.
