Question

The relation \(R\) on the set of real numbers defined by \[ R=\{(a,b):a\leq b^2\} \] is:
• [(A)] Reflexive
• [(B)] Not symmetric
• [(C)] Neither reflexive nor transitive
• [(D)] Transitive Choose the correct answer from the options given below:

• A. \((A)\ \text{and}\ (D)\ \text{only}\)
• B. \((A),\ (B)\ \text{and}\ (D)\ \text{only}\)
• C. \((B)\ \text{and}\ (C)\ \text{only}\)
• D. \((A)\ \text{and}\ (C)\ \text{only}\)

Hint:

To check properties of relations: \[ \text{Use counterexamples whenever possible.} \]

Answer 1

The Correct Option is C

Step 1: Check reflexive property A relation is reflexive if: \[ (a,a)\in R \quad \forall a\in\mathbb R \] Given: \[ a\leq a^2 \] Consider: \[ a=\frac12 \] Then: \[ \frac12 \nleq \frac14 \] which is false. Hence relation is: \[ \text{not reflexive} \] Thus: \[ (A)\ \text{is false} \]
Step 2: Check symmetric property A relation is symmetric if: \[ (a,b)\in R \Rightarrow (b,a)\in R \] Take: \[ a=1,\quad b=2 \] Then: \[ 1\leq2^2 \] so: \[ (1,2)\in R \] Now check: \[ 2\leq1^2 \] which is false. Hence relation is: \[ \text{not symmetric} \] Thus: \[ (B)\ \text{is true} \]
Step 3: Check transitive property A relation is transitive if: \[ (a,b)\in R \text{ and } (b,c)\in R \Rightarrow (a,c)\in R \] Take: \[ a=4,\quad b=3,\quad c=2 \] Then: \[ 4\leq3^2=9 \] and: \[ 3\leq2^2=4 \] So: \[ (a,b)\in R,\quad (b,c)\in R \] But: \[ 4\leq2^2=4 \] This is true, so choose another example. Take: \[ a=10,\quad b=4,\quad c=2 \] Then: \[ 10\leq4^2=16 \] and: \[ 4\leq2^2=4 \] But: \[ 10\leq2^2=4 \] which is false. Hence relation is: \[ \text{not transitive} \] Thus: \[ (D)\ \text{is false} \] Therefore relation is: \[ \text{not symmetric and neither reflexive nor transitive} \] Hence correct statements are: \[ (B)\ \text{and}\ (C) \] Therefore: \[ \boxed{\text{(C)}} \]