Let R₁ and R₂ be two relations defined on $\mathrm{ℝ}$ by ${\mathrm{aR}}_1\mathrm{b}\iff \mathrm{ab}\ge 0$ and ${\mathrm{aR}}_2\mathrm{b}\iff \mathrm{a}\ge \mathrm{b},$ then

${\mathrm{R}}_1=\{\mathrm{xy}\ge 0, \mathrm{x}, \mathrm{y}\in \mathrm{R}\}$

For reflexive $\mathrm{x}\times \mathrm{x}\ge 0$ which is true.

For symmetric

If $\mathrm{xy}\ge 0\Rightarrow \mathrm{yx}\ge 0$

If x = 2, y = 0 and z = -2

Then $\mathrm{x}.\mathrm{y}\ge 0 \mathrm{\&} \mathrm{y}.\mathrm{z}\ge 0\text{ but x.z }\ge 0$ is not true

$\Rightarrow$not transitive relation.

$\Rightarrow {\mathrm{R}}_1$ is not equivalence

${\mathrm{R}}_2\text{ if }\mathrm{a}\ge \mathrm{b}$ it does not implies $\mathrm{b}\ge \mathrm{a}$

$\Rightarrow {\mathrm{R}}_2$ is not equivalence relation

$\Rightarrow \mathrm{D}$