Question

A relation $\rho$ on the set of real number $R$ is defined as follows:
$x\rho y$ if any only if $xy>0$. 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)

The correct options are
B $\rho$ is symmetric and transitive
C $\rho$ is symmetric but not reflexive

$x\rho x:x^2>0$ not true for $x=0$
$x\rho y\Rightarrow y\rho x$
$x\rho y$ so $xy>0,y\rho z$ so $yz>0$.
$xz{y}^2>0$, Hence $xz>0$ so $x\rho z$.