(k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system of subsets of B. Define S(A) = {S(X) C B : X e A). (8) = {r-)eA :Y e B}. Prove that f-1(B) is a ring if B is a ring. (1)...

Extracted text: (k) Let f: A B be a mapping of sets, A a system of subsets of A, and B a system of subsets of B. Define S(A) = {S(X) C B : X e A). (8) = {r-)eA :Y e B}. Prove that f-1(B) is a ring if B is a ring. (1) Show that f(A) is not necessarily a ring if A is a ring. (m) Show that -(B) is a a-algebra if B is a o-algebra. (n) Prove that R(s-(8) = (R(B). where R(C) is the minimal ring containing e.