All One-One Functions from \(A=\{1,2,3\}\) to Itself
📺 Video Explanation
📝 Question
Let:
\[ A=\{1,2,3\} \]
Write all one-one functions from \(A\) to itself.
✅ Solution
Since domain and codomain both have 3 elements, every one-one function is a permutation of \(\{1,2,3\}\).
Total number of one-one functions:
\[ 3!=6 \]
🔹 All One-One Functions
1.
\[ f_1=\{(1,1),(2,2),(3,3)\} \]
2.
\[ f_2=\{(1,1),(2,3),(3,2)\} \]
3.
\[ f_3=\{(1,2),(2,1),(3,3)\} \]
4.
\[ f_4=\{(1,2),(2,3),(3,1)\} \]
5.
\[ f_5=\{(1,3),(2,1),(3,2)\} \]
6.
\[ f_6=\{(1,3),(2,2),(3,1)\} \]
🎯 Final Answer
\[ \boxed{\text{Total one-one functions }=6} \]
These are all possible one-one mappings from \(A\) to itself.
🚀 Exam Shortcut
- One-one from finite set to itself = permutations
- For 3 elements: total = \(3!=6\)
- List all arrangements systematically