Here's the question:
Verify that any equivalence relation between elements of a set makes it possible to represent the set as a union of usually disjoint equivalence classes of elements.
(It's from Zorich's Mathematical Analysis, the last question in a set of questions about fibers. I'm not sure how this question is related to fibers, and to be honest I don't understand fibers that much. )
Thank you for any help.
Well, let $R$ be an equivalence relation on a set $A$.
For each $a\in A$, let $\bar a=\{b\in A\mid aRb\}$ be the equivalence class of $a$. These classes are also called fibers.
Then the set of equivalence classes $\bar A = \{\bar a\mid a\in A\}$ forms a partition of $A$, i.e., each set is nonempty, the union of sets is $A$, and any two distinct sets are disjoint.
This is what is meant with the representation of $A$ as a disjoint union of fibers.
