Let f: N - 1 N be defined such that f(n) is the number of distinct prime factors of n . Let h: N - 1 N be a function defined by h(n) = n - f(n) . Then the function h is

Step-by-step solution

Given h(n) = n - f(n) , where f(n) is the number of distinct prime factors of n . Let us evaluate h(n) for some initial values of n 2 : h(2) = 2 - 1 = 1 h(3) = 3 - 1 = 2 h(4) = 4 - 1 = 3 h(5) = 5 - 1 = 4 h(6) = 6 - 2 = 4 Since h(5) = h(6) = 4 , the function h is not one-one. To check if h is onto, let us see if there exists any n N - 1 such that h(n) = 5 . n - f(n) = 5 n = f(n) + 5 If f(n) = 1 , then n = 6 , but f(6) = 2 1 . If f(n) = 2 , then n = 7 , but f(7) = 1 2 . If f(n) = 3 , then n = 8 , but f(8) = 1 3 . For