CAT 2000 — QA Question 50

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(f(1)) + 1

∴ 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).