Maximum Number of Equivalence Relations on Set \( A=\{1,2,3\} \)
📺 Video Explanation
📝 Question
Find the maximum number of equivalence relations on the set:
\[ A=\{1,2,3\} \]
- A. 1
- B. 2
- C. 3
- D. 5
✅ Solution
Number of equivalence relations on a finite set = number of partitions of that set.
For set with 3 elements, this is Bell Number:
\[ B_3=5 \]
🔹 Possible Partitions of \( \{1,2,3\} \)
- \(\{\{1\},\{2\},\{3\}\}\)
- \(\{\{1,2\},\{3\}\}\)
- \(\{\{1,3\},\{2\}\}\)
- \(\{\{2,3\},\{1\}\}\)
- \(\{\{1,2,3\}\}\)
Total:
\[ 5 \]
🎯 Final Answer
\[ \boxed{5} \]
✔ Correct option: D
🚀 Exam Shortcut
- Equivalence relations correspond to partitions
- For 3 elements, Bell number = 5
- Remember: \(B_1=1,\ B_2=2,\ B_3=5\)