How to describe an equivalence class?

Question

For example: the relation given is $x\sim y$ if $f(x)=f(y)$.

What do you have to say when describing a equivalence class?

Answer 1

If $f:A\to B$ denotes a function and $\sim$ is a relation on $A$ prescribed by: $$x\sim y\iff f(x)=f(y)$$ then the equivalence class represented by $a\in A$ is the set: $$f^{-1}(\{f(a)\}):=\{x\in A\mid f(x)=f(a)\}$$ You can say that $[a]_{\sim}$ is actually the fiber of $f(a)\in B$.