Basic mapping reductions without using Turing machines I have problems with the basics of mapping reductions.

I can understand how to do reductions using a Turing machine, but without it, I get a little bit confused.

For example:


How do I do a mapping reduction from $a$ to $b$, where $a$ is $\{0,1\}$ and $b$ is $\{0\}$ over the alphabet of $\{0,1\}$?
Does every language have a reduction to itself?
The identity function is a reduction from a language to itself. One reduction from $\{0,1\}$ to $\{0,1\}$ is given by $\lfloor x/2 \rfloor$, which maps $0,1$ to $0$, and all other integers to non-zero values.