Question

Let $R_1$ and $R_2$ be two relations defined on $R$ by $a{R}_{1}b\iff ab\ge 0$ and $a{R}_{2}b\iff a\ge b$, 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)

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