On the set R of real numbers, the relation rh | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) Given the relation $\rho$ on the set $R$ of real numbers defined by $x \rho y \iff x > |y|$.$1$. For reflexive property:Check if $x \rho x$ holds

Vedclass · Accepted Answer

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

Vedclass · Answer

(D) Given the relation $\rho$ on the set $R$ of real numbers defined by $x \rho y \iff x > |y|$.$1$. For reflexive property:Check if $x \rho x$ holds for all $x \in R$.$x \rho x \iff x > |x|$.This is false for all $x \leq 0$ (e.g.,if $x = -1$, $-1 > |-1| = 1$ is false).Thus, $\rho$ is not reflexive.$2$. For symmetric property:Check if $x \rho y \implies y \rho x$.$x \rho y \implies x > |y|$.$y \rho x \implies y > |x|$.If we take $x = 2$ and $y = 1$, $2 > |1|$ is true, but $1 > |2|$ is false.Thus, $\rho$ is not symmetric.$3$. For transitive property:Check if $x \rho y$ and $y \rho z \implies x \rho z$.$x \rho y \implies x > |y|$.$y \rho z \implies y > |z|$.Since $y > |z|$, we have $|y| \geq y > |z|$, so $|y| > |z|$.Since $x > |y|$ and $|y| > |z|$, by transitivity of inequality, $x > |z|$.Therefore, $x \rho z$ holds.Thus, $\rho$ is transitive.Conclusion: The relation $\rho$ defined by $x > |y|$ is transitive but neither reflexive nor symmetric.

On the set $R$ of real numbers, the relation $\rho$ is defined by $x \rho y$ if $x > |y|$. Which of the following statements is true regarding the properties of $\rho$?

Similar Questions

The relation "is subset of" on the power set $P(A)$ of a set $A$ is

On the set of all real numbers,a relation $R$ is defined as $a \, R \, b$ if and only if $|a - b| \le 1$. Then $R$ is:

Let $A = \{a, b, c\}$. The number of equivalence relations on $A$ containing $(b, c)$ is:

Let $R$ be a reflexive relation on a finite set $A$ containing $n$ elements,and let $R$ contain $m$ ordered pairs. Then,

Vedclass Test Series

Exam Paper Generator

Online Exam Module