f:\mathbb{N}\to S

**dwsmith** · June 13th 2011, 04:04 PM

$f:\mathbb{N}\to S$

Where $S:\{s\in S| \ s=2p+1, \ p\in\mathbb{Z}\}$

I need to define a 1-1 function.

$f(n)=\begin{cases}n, & \text{when n is odd}\\?, & \text{when n is even}\end{cases}$

I haven't been able to think of something that will produce negative odd integers from even natural numbers.

**Also sprach Zarathustra** · June 13th 2011, 05:47 PM

Better, one-to-one and onto function from $\mathbb{N}$ to $S$ :

$f(n)=\frac{(-1)^n}{2}(-2n+(-1)^n+1)$

Therefor, $|S|=\aleph_0$

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