In another question I had asked about notation in graph theory to identify different classes of vertices in a graph.
One of the answers gives the following example, which I'm trying to understand more fully.
Let $\mathcal{V}$ be a set of (disjoint) sets we will use for vertices of a graph $G = (V, E)$. Define $L: V \to \mathcal{V}$ by setting $L(v) = S$ when $v \in S \in \mathcal{V}$.
I'm not familiar with logic / set theory / graph theory notation, but can parse (and translate into natural language) some of it, but not all of it. I'm having trouble translating
Define $L : V \to \mathcal{V}$ by setting $L(v) = S$
Is the following "translation" correct?
$L$ is a list function of $V$ to $\mathcal{V}$, which "classifies" a vertex $v$ (the $L(v)$ part) as an element of the set $S$ (which is a set of vertices of the same type ("class")).
Or is $L$ a list of (classes of) elements of $V$, and $S$ is a specific class set (i.e., a set in $\mathcal{V}$)?
I try to avoid the word "list" when it comes to functions because that term can have mental baggage.
All that line is saying is that $L$ is a function which takes as input any vertex and outputs the class to which that vertex belongs.
It's analogous to defining $f : \mathbb{Z} \to \{\text{odd}, \text{even}\}$ as the function that takes any integer and outputs whether it's even or odd.