Question

Let $R_1$ and $R_2$ be two relations defined on $R$ by $a R _1 b \Leftrightarrow a b \geq 0$ and $a R_2 b \Leftrightarrow a \geq b$, then

• A. $R_1$ is an equivalence relation but not $R_2$
• B. $R_2$ is an equivalence relation but not $R_1$
• C. both $R_1$ and $R_2$ are equivalence relations
• D. neither $R_1$ nor $R_2$ is an equivalence relation

Answer 1

Answer: (D)

$R_1=\{x y \geq 0, x, y \in R\}$
For reflexive $x \times x \geq 0$ which is true.
For symmetric
If $x y \geq 0 \Rightarrow y x \geq 0$
If $x=2, y=0$ and $z=-2$
Then $x \cdot y \geq 0 \& y \cdot z \geq 0$ but $x . z \geq 0$ is not true
$\Rightarrow$ not transitive relation.
$\Rightarrow R_1$ is not equivalence
$R _2$ if $a \geq b$ it does not implies $b \geq a$
$\Rightarrow R_2$ is not equivalence relation
$\Rightarrow D$