Functions which map $w_n : S^n\to Z, w_n(x) =$ the number of $1$'s in $x$

Question

So, I'm having trouble understanding exactly what this function is mapping onto what. Does it map $S$, which can be either $1$ or $0$, raised to the power of $n$ to an integer?

Answer 1

$S^n=S\times S\times S...\times S$ where the cartesian product is taken $n$ times. Hence $S^n:=\{(a_1,a_2,...,a_n):a_i\in S=\{0,1\}~\forall i\}$, is the set of $n-$bit strings (bit strings of length $n$). $w_n$ maps each bit string to the number of $1$s it contains. The domain of $w_n$ is $S^n$ and the codomain is the set of integers. The range is $\Bbb Z_{\ge0}$.