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)?

CAT 2000 Question Paper Question 155

Question 155

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)?

Solution

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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Comments

Saurabh Karbhajan

2 years, 9 months ago

Condition does not satisfy for rest of the elements

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