Question

In the set of real numbers, the relation \( R \) defined by \( R = \{(a, b) : a \leq b^2 \} \) is:

• A. not reflexive and symmetric, but transitive
• B. not reflexive and transitive, but symmetric
• C. not symmetric and transitive, but reflexive
• D. not reflexive, not symmetric and not transitive

Hint:

When analyzing relations, carefully examine reflexivity, symmetry, and transitivity using the definitions provided to determine the correct classification.

Answer 1

The Correct Option is A

Step 1: Check if the relation is reflexive: For reflexivity, \( a \leq a^2 \) must hold for all real numbers. This does not hold for all values of \( a \) (e.g., for \( a = -1 \), \( -1 \leq (-1)^2 \) is false), so the relation is not reflexive. 
Step 2: Check if the relation is symmetric: The relation is not symmetric because if \( a \leq b^2 \), it does not imply that \( b \leq a^2 \) in general. For example, if \( a = 2 \) and \( b = 1 \), then \( 2 \leq 1^2 \) holds, but \( 1 \leq 2^2 \) does not hold. 
Step 3: Check if the relation is transitive: The relation is transitive. If \( a \leq b^2 \) and \( b \leq c^2 \), then we can prove \( a \leq c^2 \) holds. Therefore, the correct answer is (A) not reflexive and symmetric, but transitive.