Which Relation is a Function?
📝 Question
Let:
\[ A=\{1,2,3\} \]
Given relations:
\[ f=\{(1,3),(2,3),(3,2)\} \]
\[ g=\{(1,2),(1,3),(3,1)\} \]
Determine which of these is a function.
✅ Solution
🔹 Step 1: Definition of function
A relation is a function if every element of domain has exactly one image.
—🔹 Step 2: Check relation \(f\)
\[ 1 \to 3,\quad 2 \to 3,\quad 3 \to 2 \]
Each element of \(A\) has exactly one image.
Hence, \(f\) is a function.
—🔹 Step 3: Check relation \(g\)
\[ 1 \to 2,\quad 1 \to 3 \]
Element 1 has two images.
Hence, \(g\) is not a function.
—🎯 Final Answer
\[ \boxed{f \text{ is a function, but } g \text{ is not a function}} \]
🚀 Exam Shortcut
- Each input must have only one output
- Check first elements (domain)
- If any repeats with different outputs ⇒ not a function