One version of Ackermann's function A(m,n) is defined recursively for m, n ∈ N by
A(0, n) = n + 1, n > 0;
A(m, 0) = A(m - 1, 1), m > 0; and
A(m, n) = A(m - 1, A(m, n - 1)), m, n > 0.
[Such functions were defined in the 1920s by the German mathematician and logician Wilhelm Ackermann (1896-1962), who was a student of David Hilbert (1862-1943). These functions play an important role in computer science - in the theory of recursive functions and in the analysis of algorithms that involve the union of sets.]
(a) Calculate A(l, 3) and A(2, 3).
(b) Prove that A(l, n) = n + 2 for all n ∈ N.
(c) For all n e N show that A (2, n) = 3 + 2n.
(d) Verify that A(3, n) = 2n+3 - 3 for all n ∈ N.