Check One-One and Onto

🎥 Video Explanation

---

📝 Question

Let \( f:\mathbb{N} \to \mathbb{Z} \) be defined by:

\[ f(n)= \begin{cases} \dfrac{n-1}{2}, & \text{if } n \text{ is odd} \\ -\dfrac{n}{2}, & \text{if } n \text{ is even} \end{cases} \]

• A. neither one-one nor onto
• B. one-one but not onto
• C. onto but not one-one
• D. one-one and onto both

---

✅ Solution

🔹 Step 1: Rewrite Using \(n\)

For odd \(n=2k+1\):

\[ f(n)=\frac{2k+1-1}{2}=k \]

For even \(n=2k\):

\[ f(n)=-\frac{2k}{2}=-k \] —

🔹 Step 2: Output Pattern

\[ 0, -1, 1, -2, 2, -3, 3, \dots \]

—

🔹 Step 3: Check Injective

Each \(n\) gives a unique integer.

✔️ One-one

—

🔹 Step 4: Check Onto

Every integer appears:

• 0 from \(n=1\)
• Positive integers from odd \(n\)
• Negative integers from even \(n\)

✔️ Onto

—

🔹 Final Answer

\[ \boxed{\text{Option D: one-one and onto}} \]