Construct Mappings from \(A\) to \(B\)
📺 Video Explanation
📝 Question
Given:
\[ A=\{2,3,4\},\quad B=\{2,5,6,7\} \]
Construct examples of:
- (i) an injective map from \(A\) to \(B\)
- (ii) a mapping from \(A\) to \(B\) which is not injective
- (iii) a mapping from \(A\) to \(B\)
✅ Solution
🔹 (i) Injective Map
Choose distinct images for distinct elements:
\[ f=\{(2,2),(3,5),(4,6)\} \]
✔ This is injective.
🔹 (ii) Not Injective Map
Choose same image for two different elements:
\[ g=\{(2,2),(3,2),(4,5)\} \]
Since:
\[ g(2)=g(3)=2 \]
❌ Not injective.
🔹 (iii) Any Mapping from \(A\) to \(B\)
Example:
\[ h=\{(2,7),(3,5),(4,2)\} \]
✔ Valid mapping.
🎯 Final Answer
Examples:
(i) \[ \{(2,2),(3,5),(4,6)\} \]
(ii) \[ \{(2,2),(3,2),(4,5)\} \]
(iii) \[ \{(2,7),(3,5),(4,2)\} \]
🚀 Exam Shortcut
- Injective: all images different
- Not injective: repeat one output
- Function: every input must have one output