For example
$$f:\mathbb Z\rightarrow\mathbb N$$
$$ f(n)=\begin{cases} 2n+1 \text{ if } n\geq0 \\ -2n \text{ if } n&lt;0\end{cases}$$

How to show the integers have same cardinality as the natural numbers?
I've got a server with a LOT of small files -- many millions files, and over 1.5 TB of data. I need a decent backup strategy.


Any filesystem-based backup takes too long -- just enumerating which files need to be copied takes a day. 
Acronis can do a disk image in 24 hours, but fails when it tries to do a differential backup the next day. 
DFS-R won't replicate a volume with this many files.


I'm starting to look at Double Take, which seems to be able to do continuous replication. Are there other solutions that can do continuous replication at a block or sector level -- not file-by-file over a WAN?

Some details:


The files are split up into about 75,000 directories.
99% of the daily change comes from adding new directories; existing files are rarely changed.


There's some other relevant discussion here.
