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 $|x|>y$ then $\rho$ is reflexive and transitive but not symmetric
• D. if x > |y| then $\rho$ is transitive but neither reflexive nor symmetric

Answer 1

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

$(x,x)\in R\Rightarrow x>|x|$ false
not reflexive
$(x,y)\in R\Rightarrow x>|y|\text{/}\Rightarrow y>|x|$
$\therefore$ not symmetric
$(x,y)\in R\Rightarrow x>|y|,(y,z)\in R\Rightarrow y>|z|$
$\Rightarrow x>|z|\Rightarrow (x,z)\in R$
$\therefore$ Transitive