Let R _ 1 and R _ 2 be two relations defined on ℝ by a R _ 1 b ⇔ ab...

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 $ab \ge 0 ⇒ ba \ge 0$ which is true,

If $a = 2, b = 0$ and $c = - 2$

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

$⇒$ not transitive relation

$⇒ {R}_{1}$ is not equivalence

For ${R}_{2}$ if $a \ge b$ it does not implies $b \ge a$

$⇒ {R}_{2}$ is not equivalence relation

So, neither ${R}_{1}$ nor ${R}_{2}$ is an equivalence relation