Question

Relation \( R = \{(x, y): x<y^2 \text{ where } x, y \in \mathbb{R}\} \) is:

• A. Reflexive but not symmetric
• B. Symmetric and transitive but not reflexive
• C. Reflexive and Symmetric
• D. Neither reflexive nor symmetric nor transitive

Hint:

When checking properties of relations, remember that reflexivity, symmetry, and transitivity must hold for all elements involved in the relation.

Answer 1

The Correct Option is D

Step 1: Checking Reflexivity.
For the relation to be reflexive, \( x<x^2 \) must hold for all \( x \in \mathbb{R} \). However, for some values of \( x \), such as \( x = 0 \), this condition is not satisfied. Therefore, the relation is not reflexive. Step 2: Checking Symmetry.
For symmetry, if \( (x, y) \in R \), then \( (y, x) \) must also be in \( R \). But if \( x<y^2 \), it is not necessarily true that \( y<x^2 \). Therefore, the relation is not symmetric. Step 3: Checking Transitivity.
For transitivity, if \( (x, y) \in R \) and \( (y, z) \in R \), then \( (x, z) \) must also be in \( R \). This does not hold for all values of \( x \), \( y \), and \( z \), so the relation is not transitive. Step 4: Conclusion.
Thus, the correct answer is (D) Neither reflexive nor symmetric nor transitive.