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:

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

This means two numbers $a$ and $b$ are related if $a$ is less than or equal to $b$ squared.

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For a relation to be reflexive, $(a, a)$ must be in $R$ for all real numbers $a$.

For $(a, a)$ to be in $R$: $a \leq a^2$

Testing with $a = 0.5$:

$a = 0.5$

$a^2 = (0.5)^2 = 0.25$

Is $0.5 \leq 0.25$? No.

So $(0.5, 0.5)$ is not in $R$.

$R$ is not reflexive. Statement (A) is false.

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For a relation to be symmetric, if $(a, b)$ is in $R$, then $(b, a)$ must also be in $R$.

Testing with $a = 1, b = 2$:

Check $(1, 2)$: Is $1 \leq 2^2$? Is $1 \leq 4$? Yes.

So $(1, 2)$ is in $R$.

Check $(2, 1)$: Is $2 \leq 1^2$? Is $2 \leq 1$? No.

So $(2, 1)$ is not in $R$.

$R$ is not symmetric. Statement (B) is true.

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For a relation to be transitive, if $(a, b)$ is in $R$ and $(b, c)$ is in $R$, then $(a, c)$ must also be in $R$.

Testing with $a = 3, b = 2, c = 1.5$:

Check $(3, 2)$: Is $3 \leq 2^2$? Is $3 \leq 4$? Yes.

Check $(2, 1.5)$: Is $2 \leq 1.5^2$? Is $2 \leq 2.25$? Yes.

Check $(3, 1.5)$: Is $3 \leq 1.5^2$? Is $3 \leq 2.25$? No.

Even though $(3, 2)$ and $(2, 1.5)$ are both in $R$, $(3, 1.5)$ is not in $R$.

$R$ is not transitive. Statement (D) is false.

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Since $R$ is not reflexive and not transitive, statement (C) "Neither reflexive nor transitive" is true.

The true statements are (B) and (C).

Therefore, the correct answer is Option 3.