How interpret the notation $f:\{0,\dots, N-1\} \rightarrow \{0,\dots, N-1\}$, $N$ is a number of the form $2^n$? I need help how to interpret the following notation for $f$:


  Zeroes and ones form a binary number which can be converted to
  decimal notation. Thus, we may think of the computer as calculating a function
  $$
f:\{0,\dots, N-1\} \rightarrow \{0,\dots, N-1\}, 
$$
  where $N$ is a number of the form $2^n$, and $n$ is the number of bits in the computer memory. In this description, $f$ must be a function because the computer cannot generate two or more different outputs from the same input. We assume without loss of generality that the domain and codomain of $f$ are of the same size. In other words, we assume that both the input and the output of the computer have the same number of bits.


Update:

I understand the function notation 
\begin{align}
f&amp;:\mathbb R \rightarrow \mathbb R_+ \\
x&amp; \mapsto f(x)
\end{align}
so if $x\in \mathbb R$ we have $f(x)\in\mathbb R_+$.
So far so good.

However I don't follow the meaning (mapping) of $\{0, \dots, N-1\}$ in this case.

Attempt 1:

Say I have the decimal number $5$, so $N= 5$. I guess "of the form $2^n$" means a binary number, i.e. $5_{10}=(0101)_2$ and thus $n=4$. So I have the function
$$
f: \{0, 1, 2, 3, 4\} \rightarrow \{0, 1, 2, 3, 4\}
$$

Is this correct?

Attempt 2:

Is the domain and codomain of $f$ a binary number?  I.e. no commas in the sets
$$
f: \{0101\} \rightarrow \{0101\}
$$

Thanks in advance!
A function $f\colon \{0,\ldots,N-1\} \to \{0,\ldots,N-1\}$ accepts an input from $\{0,\ldots,N-1\}$ and outputs something from $\{0,\ldots,N-1\}$. Just like any function $f\colon A \to B$.

In this case, we are interested in $N$ of the form $2^n$. That is, we are interested in values of $N$ such that $N = 2^n$. For example, $N = 256$. We identify the set $\{0,\ldots,N-1\}$ with the set of binary strings of length $n$, using binary notation. For example, if $N = 8$, then the interpretation works as follows:


0 is interpreted as 000.
1 is interpreted as 001.
2 is interpreted as 010.
3 is interpreted as 011.
4 is interpreted as 100.
5 is interpreted as 101.
6 is interpreted as 110.
7 is interpreted as 111.


As an example, the complementation function $f\colon \{0,\ldots,7\} \to \{0,\ldots,7\}$ is given by $f(x) = 7-x$. You can check that it corresponds to complementing the binary representation of the input.