Equivalence relation and a function

Question

Suppose $A$ is a nonempty set and $R$ is an equivalence relation on $A$ . Show that there is a function $f$ with $A$ as its domain such that $(x,y) \in R$ if and only if $f(x)=f(y)$

Answer 1

Let $\equiv$ be an equivalence relation on the set $A$. Let $\bar a = \{b\in A\mid a\equiv b\}$ be the equivalence class of $a\in A$. Take the quotient set $\bar A = \{\bar a\mid a\in A\}$. Consider the mapping $f:A\rightarrow\bar A:a\mapsto \bar a$ sending each element of $A$ to its equivalence class. Then for all $a,b\in A$, $a\equiv b$ iff $\bar a = \bar b$ iff $f(a)=f(b)$.