Question

If $f : \mathbb{N} \rightarrow \mathbb{W}$ is defined as \[ f(n) = \begin{cases} \frac{n}{2}, & \text{if } n \text{ is even} \\ 0, & \text{if } n \text{ is odd} \end{cases} \] then $f$ is :

• A. injective only
• B. surjective only
• C. a bijection
• D. neither surjective nor injective

Hint:

To check if a function is injective, verify if distinct inputs map to distinct outputs. To check for surjectivity, ensure every element in the codomain has a pre-image in the domain.

Answer 1

The Correct Option is D

- Injectivity requires each domain element to map to a distinct codomain element. The function $f$ maps both even and odd numbers to different values, thus it is not injective. - Surjectivity requires every codomain element to be mapped from a domain element. However, $f$ does not map to all elements in the codomain $\mathbb{W}$; specifically, it cannot map to all odd numbers. Therefore, $f$ is not surjective. Consequently, $f$ is neither injective nor surjective.