Smallest Reflexive Relation on Set \( A = \{1,2,3,4\} \)
📺 Video Explanation
📝 Question
Write the smallest reflexive relation on the set:
\[ A = \{1,2,3,4\} \]
✅ Solution
🔹 Definition
A relation \( R \) on set \( A \) is reflexive if:
\[ (a,a) \in R \quad \forall a \in A \]
The smallest reflexive relation contains only these necessary pairs.
🔹 Construct the Relation
For each element in \( A \):
- \( (1,1) \)
- \( (2,2) \)
- \( (3,3) \)
- \( (4,4) \)
So, the smallest reflexive relation is:
\[ R = \{(1,1), (2,2), (3,3), (4,4)\} \]
🎯 Final Answer
\[ \boxed{R = \{(1,1), (2,2), (3,3), (4,4)\}} \]
🚀 Exam Insight
- Smallest reflexive relation = identity relation
- Contains only (a,a) pairs
- Number of pairs = number of elements