Let R 1 and R 2 be two relations defined on ℝ by a R 1 b ⇔ a b ≥ 0 and a   R 2 b…

Given,

$R_1=\left\{ab\ge 0,a,b\in R\right\}$

For reflexive $a\times a\ge 0$ which is true,

For symmetric

If $ab\ge 0\Rightarrow ba\ge 0$ which is true,

If $a=2,b=0$ and $c=-2$

Then $a·b\ge 0$ and $b·c\ge 0$ but $a·c\ge 0$ is not true

$\Rightarrow$ not transitive relation

$\Rightarrow R_1$ is not equivalence

For $R_2$ if $a\ge b$ it does not implies $b\ge a$

$\Rightarrow R_2$ is not equivalence relation

So, neither $R_1$ nor $R_2$ is an equivalence relation