Exercise 1.5.11 (SchroderBernstein Theorem). Assume there exists a 11 function f : X > Y and another 11 function g : Y > X. Follow the steps to show that there exists a 11, onto function h : X > Y and hence X N Y. The strategyr is to partition X and Y into components X:AUA' and YzBLJB' with A HA' 2 and B H B' = iii, in such a way that f maps A onto B, and 9 maps 3' onto A'. (a) Explain how achieving this would lead to a proof that X N Y. (b) Set A1 : X\\g(Y) : {:13 E X : m (f g[Y)} [what happens if A1 : iii?) and inductively dene a sequence of sets by letting An+1 = g(f(An]j. Show that {An : n E N} is a pairwise disjoint collection of subsets of X , while {An} : n E N} is a similar collection in Y. (e) Let A = [.1211 An and B : US$21 rst\"). Show that f maps A onto B. (d) Let A' : X\\A and B' : Y\\B. Show 9' maps 3' onto 14'