Let f be a surjective function from A into B. Define a relation R in A by setting:

(a,a) R if and only if f(a) = f(a).

(a) Show that R is an equivalence relation.

(b) Define a surjective function on the quotient set A/R by [a]R = f(a), where

[a]R denotes the equivalence class of a A, and verify this function is well-defined

(i.e., If [a] = [a] , then [a] = [a] .) RRRR

(c) Let : A A/R be the surjective function defined by (a) = [a]R. Show that = f.