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

Answer 1

Answer: (D)

If $x > |y|$ then $\rho$ is transitive but neither reflexive nor symmetric

Answer 2

On the set $R$ of real numbers For reflexive,
$x \rho x \Rightarrow(x, x) \in R$
$\Rightarrow x > |x|$ which is not true.
$\Rightarrow \rho$ is not reflexive.
For symmetric,
$(x, y) \in R \Rightarrow x > |y|$
and $(y, x) \in R \Rightarrow y > |x|$
So, $x>|y| \neq y > |x|$
$\Rightarrow \rho$ is not symmetric.
For transitive,
$(x, y) \in R \Rightarrow x>|y | (y, z) \in R \Rightarrow y>| z \mid$
$\Rightarrow x>|z| \Rightarrow(x, z) \in R$
$\Rightarrow \rho$ is transitive.