I'm trying to solve exercise 1.3 in Aluffi's book 'Chapter Zero':
Given a partition $\mathscr{P}$ on a set $S$, show how to define a relation on $S$ such that $\mathscr{P}$ is the corresponding partition.
Here is my attempt.
Let $\mathscr{P} = \{X_{\alpha}\}_{\alpha \in I}$, where $I$ is an arbitrary indexing set and $X_i \cap X_j = \emptyset$ for any $i \neq j$ in $I$. (That is, $S = \bigsqcup\limits_{\alpha \in I} X_{\alpha}$.) Define the equivalence relation $\sim$ on $S$ by $a \sim b$ if and only if $a,b \in X_{\alpha}$ for some (unique) $\alpha \in I$. It is easy to see that $\sim$ is an equivalence relation. Certainly $a \sim a$ since $a$ is in the same set $X_{\alpha}$ as itself; since the $X_{\alpha}$ partition $S$, there exists such an $X_{\alpha}$, and it's unique since the sets in the partition are disjoint.) If $a \sim b$, then $a$ and $b$ live in the same $X_{\alpha}$, so $b$ and $a$ certainly live in $X_{\alpha}$. Finally, if $a,b \in X_{\alpha}$ and $b,c \in X_{\beta}$, then $b \in X_{\alpha} \cap X_{\beta}$, so $X_{\alpha} = X_{\beta}$ since the sets are disjoint, so we have $a \sim c$. The partition into equivalence classes is exactly $\mathscr{P}$ by construction.
How does this look? I. worry my answer isn't sufficiently rigorous.