(Solved): please do both 29 and 30 since they are related to each other. please show clear steps with clear ha ...

29. Let R be a ring, and let RR be the set of all functions mapping Rinto R. For ø, VER®, define the sum $ + by (6 + V)(r) = $(r) + (1) and the producto. V by (0.1)) = o(r)y(r) for r e R. Note that is not function composition. Show that (R", +. ) is a ring. 30. Referring to Exercise 29, let F be a field. An element o of FF is a polynomial function on F. if there exists f(x) € F[x] such that o(a) = f(a) for all a € F. a. Show that the set P, of all polynomial functions on F forms a subring of F'. b. Show that the ring PF is not necessarily isomorphic to F[x]. (Hint: Show that if F is a finite field, Pe and F[x] don't even have the same number of elements.]