Answer
Showing 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(1) = 0.
c) F(n) = 1 + F(n/3) for n ≥ 3, F(1) = 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(1) = 1.
e) F(n) = 1 + F(F(n − 1)) if n ≥ 2 and F(1) = 2.
Work Step by Step
a) The value of F(1) is ambiguous.
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b) F(2) is not defined because F(0) is not defined. c) F(3)
is ambiguous and F(4) is not defined because F(4 3 ) makes
no sense.
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d) The definition of F(1) is ambiguous because
both the second and third clause seem to apply.
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e) F(2) cannot be computed because trying to compute F(2) gives
F(2) = 1 + F(F(1)) = 1 + F(2).
