Question

A relation $\rho$ on the set of real number R is defined as $\{x\rho y:xy>0\}$. Then, which of the following is/are true?

• A. $\rho$ is reflexive and symmetric
• B. $\rho$ is symmetric but not reflexive
• C. $\rho$ is symmetric and transitive
• D. $\rho$ is an equivalence relation

Answer 1

Answer: (B), (C)

$\rho$ is symmetric but not reflexive
$\rho$ is symmetric and transitive

We have, $x\rho y:xy>0$
i) Reflexive

Suppose $x\rho x\in R$
$\Rightarrow x^2>0$
Which is not true when $x=0$.
Hence, relation is not reflexive.

ii) Symmetric $x\rho t\in R$
$\Rightarrow xy>0$
$\Rightarrow yx>0$
$\Rightarrow y\rho x\in R$
If $(x,y)\in R$, then, relation is symmetric.

iii) Transitive $x\rho y\in R$
$\Rightarrow xy>0$
and $y\rho z\in R$
$\Rightarrow yz>0$
$\Rightarrow x{y}^{2}z>0$
$\Rightarrow xz>0$
$\Rightarrow (x,z)\in R$
If $(x,y)\in R$, then $(y,z)\in R$
$\Rightarrow (x,z)\in R$
Hence, relation is transitive.