Question

Let $ R $ be a relation on set $ A $ such that $ R = R^{-1} $, then $ R $ is

• A. Reflexive
• B. Symmetric
• C. Transitive
• D. None of these

Hint:

A relation \( R \) is symmetric if for every pair \( (x, y) \), the reverse pair \( (y, x) \) is also in the relation.

Answer 1

The Correct Option is B

Step 1: Understanding the Relation
The notation \( R = R^{-1} \) means that the relation \( R \) is equal to its inverse.
In other words, for every pair \( (x, y) \) in \( R \), the pair \( (y, x) \) must also be in \( R \).
This property defines a symmetric relation.
Step 2: Explanation of Other Options
Option (a) Reflexive is incorrect because reflexivity means that \( (x, x) \) must be in \( R \) for all elements in \( A \), which is not necessarily true for \( R = R^{-1} \).
Option (c) Transitive is incorrect because transitivity means if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \), which is not implied by \( R = R^{-1} \).
Option (d) None of these is incorrect because the relation is symmetric.
Step 3: Conclusion Thus, the relation \( R = R^{-1} \) is symmetric.