Show that each of these proposed recursive definitions of a function on the set of positive integers does not produce a well-defined function.

a) F(n) = 1 + F([(n + 1)/2]) for n ≥ 1 and F(1) = 1.

b) F(n) = 1 + F(n - 2) for n ≥2 and F(l) = 0.

c) F(n) = 1 + F(n/3) for n ≥ 3, F(l) = 1, F(2) = 2, and F(3) = 3.

d) F(n) = 1 + F(n/2) if n is even and n ≥ 2,F(n) = 1 + F(n - 2) if n is odd, and F(l) = 1.

e) F(n) = 1 + F(F(n - 1)) if n ≥ 2 and F(l) = 2.

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