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.
✅ 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.
🔹 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.
🎯 Final Answer
\[ \boxed{\text{The function is neither one-one nor onto}} \]
🚀 Exam Shortcut
- Same output for two inputs → not one-one
- Missing codomain element → not onto
- Check mapping arrows carefully