Check Whether Given Function is One-One and Onto

📺 Video Explanation

📝 Question

Given:

\[ A=\{a,b,c,d\},\quad B=\{x,y,z\} \]

and function:

\[ f=\{(a,x),(b,x),(c,z),(d,z)\} \]

Check whether the function is one-one and onto.

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✅ Solution

🔹 Check One-One (Injective)

A function is one-one if different inputs have different outputs.

Here:

• \(a \mapsto x\)
• \(b \mapsto x\)

Different inputs have same output.

❌ Not one-one.

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🔹 Check Onto (Surjective)

A function is onto if every element of codomain is used.

Codomain:

\[ B=\{x,y,z\} \]

Range:

\[ \{x,z\} \]

Element:

\[ y \]

has no pre-image.

❌ Not onto.

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🎯 Final Answer

\[ \boxed{\text{The function is neither one-one nor onto}} \]

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🚀 Exam Shortcut

• Same output for two inputs → not one-one
• Missing codomain element → not onto
• Check mapping arrows carefully