Question

On the set $R$ of real numbers, the relation $\rho$ is defined by $x\rho y,(x,y)\in R$.

• A. if $|x-y|<2$ then $\rho$ is reflexive but neither symmetric nor transitive
• B. if $|x-y|<2$ then $\rho$ is reflexive and symmetric but not transitive
• C. if $\left|x\right|\ge y$ then $\rho$ is reflexive and transitive but not symmetric
• D. if $x>\left|y\right|$ then $\rho$ is transitive but neither reflexive nor symmetric

Answer 1

The correct option is D if $x>\left|y\right|$ then $\rho$ is transitive but neither reflexive nor symmetric

$(x,x)$ belongs to $R\Rightarrow$ $x>|x|$ false
$\therefore$ not reflexive
$(x,y)$ belongs to $R\Rightarrow$ $x>|y|$ and $|y|$ is not$>x$
$\therefore$ not symmetric
$(x,y)$ belongs to $R$ $\Rightarrow$ $x>|y|,(y,z)$ belongs to $R$ $\Rightarrow$ $y>|z|$
$\Rightarrow$ $x>|z|$ $\Rightarrow (x,z)$ belongs to $R$
$\therefore$Transitive