Check Injective / Surjective

🎥 Video Explanation

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📝 Question

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

\[ f(x)= \begin{cases} \dfrac{x}{2}, & \text{if } x \text{ is even} \\ 0, & \text{if } x \text{ is odd} \end{cases} \]

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

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✅ Solution

🔹 Step 1: Check Injective

For all odd \(x\), \(f(x)=0\).

Different inputs → same output ⇒ ❌ Not one-one

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🔹 Step 2: Check Surjective

For any integer \(k\), choose \(x=2k\) (even):

\[ f(2k)=k \]

Every integer is covered ⇒ ✔️ Onto

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🔹 Final Answer

\[ \boxed{\text{Option A: onto but not one-one}} \]