Relation Between \( R \) and \( R^{-1} \)
📺 Video Explanation
📝 Question
If \( R \) is a symmetric relation on a set \( A \), find the relation between \( R \) and \( R^{-1} \).
✅ Solution
🔹 Definition of Symmetric Relation
A relation \( R \) is symmetric if:
\[ (a,b) \in R \Rightarrow (b,a) \in R \]
🔹 Definition of Inverse Relation
The inverse relation \( R^{-1} \) is defined as:
\[ R^{-1} = \{(b,a) : (a,b) \in R\} \]
🔹 Relation Between \( R \) and \( R^{-1} \)
If \( (a,b) \in R \), then since \( R \) is symmetric:
\[ (b,a) \in R \]
But \( (b,a) \in R^{-1} \) by definition.
So, every element of \( R \) is also in \( R^{-1} \).
Similarly, every element of \( R^{-1} \) is in \( R \).
Therefore:
\[ R = R^{-1} \]
🎯 Final Answer
\[ \boxed{R = R^{-1}} \]
🚀 Exam Insight
- Symmetric relation ⇒ pairs appear in reverse
- Inverse just flips pairs
- So both become identical
- Key result: symmetric ⇔ \( R = R^{-1} \)