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)?
Explanation:
The functions f(n) and g(n) are disjoint sets and union of these two sets is the set of all positive integers.
∵ g(n) = f(f(n)) + 1 for all n ≥ 1
and f (1) < f(2) <...< f(n) ....., and g(1) < g(2) <...< g(n) .......,
∴ f(1) = 1 or 2
If f(1) = 1
g(1) = f(f(1)) + 1
∴ g(1) = f (1) + 1 = 1 + 1 = 2
If f(1) = 2
∴ g(1) = f (2) + 1
∴ g(1) is greater than f(1), i.e. it is greater than 2.
But the set of all positive integers is the union of these two disjoint sets.
∴ This set has to include 1 which is not possible in this case as f(1) is 2 and g(1) will be greater than f(1).
∴ f(1) = 1 and g(1) = 2
Hence, option (b).
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