I want to prove that given a partition $\lbrace A_i|i\in I\rbrace$ of $A$(where $I$ is some indexing set), that there exists some equivalence relation $\sim$ such that the equivalence classes of $\sim$ are the sets $A_i$.
Here is my attempt at the proof:
Let $\lbrace A_i|i\in I\rbrace$ be a partition of $A$. Let there be a binary relation $\sim$ on $A$ such that $\forall a,b\in A$, $a\sim b$ if $a,b\in A_i$. If $a\in A_i$, then $a\sim a$ must be true. If $a,b\in A_i$, then $b,a\in A_i$, meaning $a\sim b$, and $b\sim a$. If $a,b\in A_i$, and $b,c\in A_i$, then $a,c\in A_i$, meaning $a\sim b$, $b\sim c\implies a\sim c$. Therefore, $\sim$ is an equivalence relation. For any element $a\in A$, $[a]=\lbrace x\in A|\space x\sim a\rbrace=\lbrace x\in A|\space x\in A_i\rbrace$. Therefore, each equivalence class of $\sim$ is a unique set $A_i$.
I want to know if
$1)$ There are any corrections to be made and
$2)$ Better wording for the proof.
Thank you all in advance.