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Le the relation $\rho$ be defined on $\mathbb{R}$ by a $\rho$ b holds if and only if $a-b$ is zero or irrational, then
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Verified Answer
The correct answer is:
$\rho$ is reflexive $\&$ symmetric but is not transitive
Hint:
If $a-b=0$ then $b-a=0$, if $a-b$ is irrational then $b-a$ is irrational
$\therefore a \rho b \Rightarrow b \rho a \Rightarrow$ symmetric
$\forall a \in \mathbb{R}, a-a=0 a \rho a \Rightarrow$ reflexive
If $a=2, b=\sqrt{2}, c=3$, then
a $\rho$ b, b $\rho$ c but a $\rho$ c is not true $\Rightarrow$ not transitive
If $a-b=0$ then $b-a=0$, if $a-b$ is irrational then $b-a$ is irrational
$\therefore a \rho b \Rightarrow b \rho a \Rightarrow$ symmetric
$\forall a \in \mathbb{R}, a-a=0 a \rho a \Rightarrow$ reflexive
If $a=2, b=\sqrt{2}, c=3$, then
a $\rho$ b, b $\rho$ c but a $\rho$ c is not true $\Rightarrow$ not transitive
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