Question

Prove that the function f is surjective, where f: N → N such that

\(f(n) = \begin{cases}\frac{n+1}2&, \text{if n is odd}\\\frac n2&, \text{if n is even}\end{cases}\)

Is the function injective? Justify your answer.

Answer 1

Let y ∈ N(codomain). Then ∃ 2y ∈ N(domain) such that

f(2y) = 2y/2 = y. Hence, f is surjective.

1, 2 ∈ N(domain) such that f(1) = 1 = f(2)

Hence, f is not injective.