Question

Define a relation \( R \) on the interval \( [0, \frac{\pi}{2}] \) by \( xRy \) if and only if \( \sec^2 x - \tan^2 y = 1 \). Then \( R \) is:

• A. both reflexive and transitive but not symmetric
• B. both reflexive and symmetric but not transitive
• C. reflexive but neither symmetric nor transitive
• D. an equivalence relation

Hint:

To verify if a relation is an equivalence relation, check if it is reflexive, symmetric, and transitive.

Answer 1

The Correct Option is D

- The relation \( R \) is reflexive, symmetric, and transitive.
- Reflexive: For any \( x \), \( \sec^2 x - \tan^2 x = 1 \), so \( xRx \).
- Symmetric: If \( \sec^2 x - \tan^2 y = 1 \), then \( \sec^2 y - \tan^2 x = 1 \), so \( xRy \) implies \( yRx \).
- Transitive: If \( xRy \) and \( yRz \), then \( xRz \), as the relation holds for all pairs. 

Thus, the relation is an equivalence relation.