Need help with e and f

For (c), (d), (e) below, define the set A of real numbers expressible by radicals recursively as follows: . If x E Q, then x E A. . If x E A and x > 0, then for every n E {2, 3, . ..} the positive nth root vx is in A. . If x, y E A, then x + y and xy are in A. (a) (2 points) Let A and B be countably infinite sets and fix bijections f : N - A and g : N - B. Supposing A and B are disjoint, give in terms of f and g an explicit bijection h : N - AUB showing the latter is countably infinite. Prove that h is a bijection. (b) (2 points) Prove that the set I of irrational numbers is uncountable. (c) (1 point) Is -V2 in A? Explain why or why not. (d) (1 point) Give a few more characteristic examples of elements of A. (e) (3 points) We wish to define a function d : A - N so that, intuitively, d(x) is the largest number of nested radicals in the least complicated expression for x. So for instance d(5/3) = 0 and d( V2 +