Question

Let
f,g:N = {1} -> N be functions defined by
f(a) = α, where α is the maximum of the powers of those primes p such that p α divides a , and g(a) = a + 1, for all a ∈ N - {1}. Then, the function f + g is

• A. One-one but not onto
• B. Onto but not one-one
• C. Both one-one and onto
• D. Neither one-one nor onto

Answer 1

Answer: (D)

Neither one-one nor onto

Answer 2

The correct answer is (D):
f, g : N - {1} -> N defined as
f(a) = α, where α is the maximum power of those primes p such that p α divides a.
g(a) = a + 1,
Now, f(2) = 1, g(2) = 3 ⇒ (f + g) (2) = 4
f(3) = 1, g(3) = 4 ⇒ (f + g) (3) = 5
f(4) = 2, g(4) = 5 ⇒ (f + g) (4) = 7
f(5) = 1, g(5) = 6 ⇒ (f + g) (5) = 7
∵ (f + g) (5) = (f + g) (4)
∴ f + g is not one-one
Now, ∵ f min = 1, g min = 3
So, there does not exist any x ∈ N - {1} such that
(f + g)(x) = 1, 2, 3
∴ f + g is not onto