only P4.6.8 part a) needs to be solved. Thank you

P4.6.7 (uses Java) In this problem we recursively define two more binary operators on naturals. Each operation is defined only if the second argument is nonzero. We define R(0, y) to be 0, and define R(S(x), y) to be S(R(x, y) ) unless S(R(x, y)) = y, in which case R(Sx, y) = 0. We define Q(0, x) to be 0, and define Q(Sx, y) to be Q(x, y) unless R(Sx, y) = 0, in which case Q(Sx, y) = S(Q(x, y)). (a) Write recursive static pseudo-Java methods natural r(natural x, natural y) and natural q(natural x, natural y) to compute these two operations. (b) Compute the values Q(5, 2) and R(5, 2), either using your method or working directly with the definitions. P4.6.8 Using the definition of the operators R and Q in Problem 4.6.7, prove the following facts by induction for any fixed positive natural y. (a For any natural x, y(Q(x, y)) + R(x, y) = x. (b) For any natural z, Q(zy, y) = z and R(zy, y) = 0. (c) If x is any fixed positive natural, then for natural z, Q(Q(zxy, x), y) = Q(Q(zxy, y), x)