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

• Great. thank you sir. – AnnieOK Apr 26 '14 at 22:02

HINT: Find a countably infinite set $X$ such that this set has a natural bijection with $\mathcal P(X)$.