Assignment:
Determine whether each of these proposed definitions is a valid recursive definition of a function f from the set of non negative integers to the set of integers. If f is well defined, find a formula for f(n) when n is a non negative integer and prove that your formula is valid.
a) f(0) = 1,f(n) = - f(n - 1) for n >= 1
b) f(0) = 1, f(1) = 0, f(2) = 2, f(n) = 2f(n - 3) for n>=3
c) f(0) = 0, f(1) = 1, f(n) = 2f(n + 1) for n >= 2
d) f(0) = 0, f(1) = 1, f(n) = 2f(n - 1) for n >= 1
e) f(0) = 2, f(n) = fen - 1) if n is odd and n >= 1 and f(n) = 2f(n - 2) if n >= 2
Provide complete and step by step solution for the question and show calculations and use formulas.