The question is as follows,
Assume is onto. Define a relation of given by iff .
1) Show that every equivalence class with respect to is of the form for some element .
So, the equivalence classes of are given by
which I possibly could write as
where, . But, how can this inverse exist in the first place; he defined the map to be onto, and not one-to-one! Am I thinking about this the wrong way?
2) Show that the there is a one-to-one correspondence between and the set of equivalence classes of
My argument goes a little like this; we have our set , and denote our set by . Then, we have that each of these elements are unique. If this is the case, then each equivalence on the set corresponds to a unique element on the set . This satisfies injectivity.
I'm not sure if this is even a correct argument, or if I'm on the right tracks, and how would this all be formalised?
3) Prove that for every equivalence relation on , one can find a and an onto map such that
I have not got a clue what on earth this is even asking! Any help on this would be much appreciated.
Thank you all in advance,