Let X and Y be two sets and EC P(X), FC P(Y). Denote by M(E) and M(F) the -algebras generated respectively by E and F. Consider a map f: XY. 1. (Pre-image o-algebra) Prove that if B is a o-algebra on Y, then J- (B) := {(B), B&B}, is a o-algebra in X. 2. Prove that if A is a o-algebra on X, then := {BCY, (B)EA}. is a a-algebra on Y. Note that, in general, (4):= {(4) CY, A A}, is not a o-algebra in Y (for example if f is not surjective, Y# f(A)). 3. Let X = {a,b. c) and Y= {1,2} and f the surjective map from X to Y defined by f(x) = 1 if rc and f(c) = 2. Consider S:= {0, {a}, {b. c), X}. Prove that S is a o-algebra on X, however f(S) is not a o-algebra on Y. 4. Let C P(Y). Prove that Prove that is a o-algebra in Y. Deduce that M(f(C)) c f(M(C)). B:= {BCY, f(B) e M(f(c))}, M(f (C)) = f(M(C)).