Check One-One and Onto

🎥 Video Explanation

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📝 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

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✅ Solution

🔹 Step 1: Evaluate Pattern

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: Outputs

Function produces:

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

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🔹 Step 3: Check Injective

Each natural number maps to a unique integer.

No repetition ⇒ ✔️ one-one

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🔹 Step 4: Check Onto

Every integer \(k\) is covered:

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

✔️ onto

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🔹 Final Answer

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