Question

A relation $\rho$ on the set of real number $R$ is defined as follows :
$x\rho y$ if and only if $xy>0$. Then which of the followings 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 but not reflexive
C $\rho$ is symmetric and transitive


For reflexivity,
Let $a\in R$, then $(a,a)\notin \rho$ for $a=0$.
Thus, the relation $\rho$ is not reflexive.

For symmetricity,
Let $(a,b)\in \rho$
$\Rightarrow ab>0\Rightarrow ba>0$
Thus, $(b,a)\in \rho$.
Thus, the relation $\rho$ is symmetric.

For transitivity,
Let $(a,b)\in \rho$ and $(b,c)\in \rho$
$\Rightarrow ab>0,bc>0$
This implies $a{b}^{2}c>0\Rightarrow ac>0$
Thus, $(a,c)\in \rho$.
Thus, the relation $\rho$ is transitive.