Problem: Let be a surjective map of sets. Prove that the relation is an equivalence relation whose equivalence classes are the fibers of .
My Work/Thoughts: What does a surjective map of sets mean? I think it means that for all there is some such that . An equivalence relation has to be symmetric, reflexive, and transitive. I know that really means that which is some subset of a set . So we have to show that these 3 properties hold.
(i) reflexive: for all . So ?
(ii) symmetric: for all . So and .
(iii) transitive: and for all . So for some ? So the ' ' sign acts like the ' '?
Now I have to show that the equivalence classes are the fibers of . So by definition, an equivalence class is . This is where I become stuck. A fiber is the inverse image of one element of the codomain mapped to the domain right? So is it ?
Thanks