Equivalence Class of \( (3,2) \) in \( A \times A \)
📺 Video Explanation
📝 Question
Let \[ A=\{2,3,4,5,\dots,18\} \]
An equivalence relation \( \simeq \) on \( A\times A \) is defined by:
\[ (a,b)\simeq(c,d)\iff ad=bc \]
Find the number of ordered pairs in the equivalence class of:
\[ (3,2) \]
✅ Solution
Given:
\[ (a,b)\simeq(3,2) \]
Using relation:
\[ a\cdot 2=b\cdot 3 \]
So:
\[ 2a=3b \]
We need all pairs \((a,b)\in A\times A\) satisfying:
\[ \frac{a}{b}=\frac{3}{2} \]
🔹 General Form
Let:
\[ a=3k,\quad b=2k \]
where both must belong to:
\[ A=\{2,3,4,\dots,18\} \]
🔹 Find Possible Values of \( k \)
Conditions:
\[ 3k\leq18 \Rightarrow k\leq6 \]
\[ 2k\geq2 \Rightarrow k\geq1 \]
So:
\[ k=1,2,3,4,5,6 \]
🔹 Ordered Pairs
- \((3,2)\)
- \((6,4)\)
- \((9,6)\)
- \((12,8)\)
- \((15,10)\)
- \((18,12)\)
Total pairs:
\[ 6 \]
🎯 Final Answer
\[ \boxed{6} \]
✔ Correct option: (c)
🚀 Exam Shortcut
- Equivalence class means same ratio
- Write in simplest ratio form
- Count multiples within given set