Injection with Range \( \{a\} \)
📺 Video Explanation
📝 Question
If:
\[ f:A\to B \]
is an injection and:
\[ \text{Range}(f)=\{a\} \]
determine the number of elements in set \(A\).
✅ Solution
Given:
\[ \text{Range}(f)=\{a\} \]
This means every element of \(A\) maps to the same value \(a\).
🔹 Injection Property
For injection:
\[ f(x_1)=f(x_2)\Rightarrow x_1=x_2 \]
But here:
all outputs are same:
\[ f(x)=a \]
for every \(x\in A\).
So if there were two different elements in \(A\), they would have same image, which is not possible for injection.
🔹 Conclusion
Therefore, set \(A\) can have only one element.
🎯 Final Answer
\[ \boxed{n(A)=1} \]
🚀 Exam Shortcut
- Injection means distinct inputs → distinct outputs
- Single output means only one input possible
- So domain must have exactly one element