Quotients of sets.

Let $A$ be a set and let $\sim\subset A \times A$. We call $\sim$ a binary relation and write $a \sim b$ for $(a, b)\in \sim$. A subset $S\subset A$ is said to be closed under $\sim$ if whenever $x\sim y$ and one of $x$ and $y$ is in $S$, so is the other.

1. For $a\in A$, show that there is a (unique) set $[a]\subset A$ such that
   • $a$ is closed under $\sim$
   • if $S$ is closed under $\sim$ and $a\in S$ then $[a]\subset S$
2. Show that for $x\in A$, $x\in [a]$ if and only if $[x]=[a]$.
3. Conclude that $\bar{A} = \{S \subset A: \text{$S = [a]$ for some $a\in A$}\}$ is a collection of disjoint sets whose union is $A$.
4. Define the map $q: A\to \bar{A}$ by $q(a) = [a]$. Let $B$ be a set and $f: A \to B$ be a function. Show that the following are equivalent.
   • for all $a, b \in A$, $a\sim b \implies f(a) = f(b)$
   • there is a function $\bar{f} : \bar{A} \to B$ such that $\bar{f} \circ q = f$