$S$ is a finite set, and $v: S \to\mathbb N$ is a valuation function.
We have $$S := \{0,1,2\}$$
and the valuation function $v$
$$v: S \to\mathbb N\\ k \mapsto 2k+1$$
The task is to list all values for $v$'s utility function $u$: $$u:\mathcal P(S)\to\mathbb N\\ X\mapsto\sum_{x\in X} v(x)$$
How would I go on about doing so? We weren't introduced to the concept of valuation and utility functions yet, and the internet hasn't exactly helped me at all so far.
Thanks in advance.
Since the tasks seems rather straight-forward to me, I fear I'm missing something crucial in your question. Nevertheless, this is how I'd do it.
Reading the definitions, you must list $v(X)$ as $X$ ranges among the subsets $S$:
Pick a subset $X$ of $\{0,1,2\}$.
Calculate $u(X)=v(k_1)+\cdots+ v(k_h)=(2k_1+1)+\cdots+(2k_h+1)$, where $k_1,\cdots, k_h$ are the elements of $X$
Write down the number you got and restart from step 1 with a subset you haven't already considered.
For instance, let's do one go
We pick, say, $X=\{0,2\}$
$u(\{0,2\})=(2\cdot 0+1)+(2\cdot 2+1)=6$
Add $6$ to our list of values
And the rest are similar. Recall that $S$ has exactly $8$ subsets, one of which is $\emptyset$.