Question

Let \( f:\mathbb{N} \to \mathbb{N} \) be defined as \[ f(n)= \begin{cases} \frac{n+1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases} \] Then \( f \) is:

• A. injective but not surjective
• B. surjective but not injective
• C. both injective and surjective
• D. neither injective nor surjective

Hint:

Check injectivity using counterexample.

Answer 1

The Correct Option is B

Concept:

Step 1: Check injectivity: \[ f(2)=1,\quad f(1)=1 \Rightarrow f(1)=f(2) \] Not injective.

Step 2: Check surjectivity: For any \(k \in \mathbb{N}\), \[ f(2k)=k \Rightarrow \text{every value is achieved} \]