I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like,
Proposition: If $\{A_i|i\in I\}$ is a partition of $\mathcal A$, then there is an equivalence relation on $\mathcal A$ whose equivalence clases are precisely the sets $A_i, i \in I$.
Where $I$ is some indexing set.
How do I prove the statement ? I can't even decide on a good place to start.
When $I$ is an arbitrary "index set" then a set-vauled function $$f:\quad I\to {\cal P}(A)\ ,\quad i\mapsto A_i\ ,$$ where ${\cal P}(A)$ denotes the power set of $A$, is called a family of subsets of $A$ and is denoted by $(A_i)_{i\in I}\ $. Such a family is called a partition of $A$, if (i) all $A_i$ are nonempty, (ii) the $A_i$ are pairwise disjoint, i.e., $A_i\cap A_j=\emptyset$ when $i\ne j$, and (iii) $\bigcup_{i\in I} A_i=A$.
Given a partition $(A_i)_{i\in I}$ of $A$, for each $x\in A$ there is a unique $i\in I$ such that $x\in A_i$. This defines a function $\iota: A\to I$ which returns for each $x\in A$ the unique $i$ with $x\in A_i$.
Now it is easy to check that $$x\sim y\quad:\Leftrightarrow\quad \iota(x)=\iota(y)$$ defines an equivalence relation on $ A$ whose equivalence classes are exactly the $A_i$.