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The set of all positive integers is the union of two disjoint subsets:
{f(1), f(2),.....f(n), ...} and {g(1),g(2).... ,g(n).....}, where f(1) < f(2) <.....< f(n)..., and g(1) < g(2) < ..... < g(n) ...,and
g(n) = f(f(n))+1 for all n >= 1. What is the value of g(1)?
The union of the two sets is the set of positive integers. Also, given the increasing nature of elements, either f(1) or g(1) must be equal to 1. If g(1) = 1, then f(f(1)) =0 which cannot be under the given conditions.
Hence, f(1) = 1
g(1) = f(f(1))+1 = 2
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