Question

The relation \(R=\{(a,b): a\le b^2\}\) is defined in the set of real numbers. Then \(R\) is:

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

Hint:

Test a=0.5 for reflexivity, (0,10) vs (10,0) for symmetry, and (10,4,2) for transitivity — all fail.

Answer 1

The Correct Option is D

Step 1: Check reflexivity:
Reflexive needs \(a\le a^2\) for every real \(a\), i.e. \(a^2-a\ge0\), i.e. \(a(a-1)\ge0\), true only for \(a\le0\) or \(a\ge1\). Take \(a=0.5\): \(0.5\le0.25\) is false. So \(R\) is not reflexive.

Step 2: Check symmetry:
Take \(a=0,\ b=10\): \(0\le10^2=100\) is true, so \((0,10)\in R\). But is \((10,0)\in R\)? That needs \(10\le0^2=0\), false. So \(R\) is not symmetric.

Step 3: Check transitivity:
Take \(a=10,\ b=4,\ c=2\). \(a\le b^2\): \(10\le16\) true. \(b\le c^2\): \(4\le4\) true. But \(a\le c^2\): \(10\le4\) is false. So \(R\) is not transitive.

Final Answer:
\(R\) fails reflexivity, symmetry and transitivity all three. \[ \boxed{\text{Not reflexive, not symmetric, not transitive}} \]