Question

Define a relation \( R \) on the interval \( \left[ 0, \frac{\pi}{2} \right] \) by \( x \, R \, y \) 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 determine if a relation is an equivalence relation, check if it satisfies reflexivity, symmetry, and transitivity. If all three properties hold, it is an equivalence relation.

Answer 1

The Correct Option is D

Step 1: Recall the definition of an equivalence relation: a relation \( R \) is an equivalence relation if it satisfies reflexivity, symmetry, and transitivity.
Step 2: We verify the properties for the given relation: - Reflexive: For all \( x \) in the interval \( \left[ 0, \frac{\pi}{2} \right] \), we must confirm \( x R x \). This requires checking if \( \sec^2 x - \tan^2 x = 1 \). This identity holds for all \( x \) in the specified interval, thus the relation is reflexive. 
- Symmetric: If \( x R y \) holds, then \( y R x \) must also hold for symmetry. Given the symmetric nature of the equation involving \( x \) and \( y \), this property is satisfied. 
- Transitive: If \( x R y \) and \( y R z \) are true, then \( x R z \) must also be true for transitivity. This property is also satisfied, confirming the relation is transitive. Therefore, since \( R \) is reflexive, symmetric, and transitive, it is an equivalence relation.