Let a relation $\mathrm{R}$ on $\mathrm{N} \times \mathbb{N}$ be defined as: $\left(x_1, y_1\right) \mathrm{R}\left(x_2, y_2\right)$ if and only if $x_1 \leq x_2$ or $y_1 \leq y_2$. Consider the two statements:

(I) $\mathrm{R}$ is reflexive but not symmetric.

(II) $\mathrm{R}$ is transitive

Then which one of the following is true?

<p>$$\begin{aligned} & \left(x_1, y_1\right) R\left(x_2, y_2\right) \\ & \text { If } x_1 \leq x_2 \text { or } y_1 \leq y_2 \end{aligned}$$</p> <p>For reflexive;</p> <p>$$\begin{aligned} & \left(x_1, y_1\right) R\left(x_1, y_1\right) \\ & \Rightarrow x_1 \leq x_1 \text { or } y_1 \leq y_1 \end{aligned}$$</p> <p>So, $R$ is reflexive</p> <p>For symmetric</p> <p>When $\left(x_1, y_1\right) R\left(x_2, y_2\right)$</p> <p>$\Rightarrow x_1 \leq x_2 \text { or } y_1 \leq y_2$</p> <p>For $\left(x_2, y_2\right) R\left(x_1, y_1\right)$</p> <p>$\Rightarrow x_2 \leq x_1 \text { or } y_2 \leq y_1$</p> <p>Not true for $(1,2)$ and $(3,4)$</p> <p>For transitive</p> <p>Take pairs as $(3,9),(4,6),(2,7)$</p> <p>$(3,9) R(4,6)$</p> <p>as $4 \geq 3$</p> <p>$(4,6) R(2,7)$</p> <p>As $7 \geq 6$</p> <p>But $(3,9) R(2,7)$</p> <p>As neither $2 \geq 3$ nor $7 \geq 9$</p> <p>So not transitive</p>