A relation rho on the set of real numbers mat | vedclass

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(C) We have the relation $\rho$ defined as $x \rho y \iff xy > 0$.$(i)$ Reflexive: For $\rho$ to be reflexive, $x \rho x$ must hold for all $x \in \ma

Vedclass · Accepted Answer

$\rho$ is symmetric and transitive

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(C) We have the relation $ho$ defined as $x ho y \iff xy > 0$.$(i)$ Reflexive: For $ho$ to be reflexive, $x ho x$ must hold for all $x \in \mathbb{R}$. This implies $x \cdot x > 0$, or $x^2 > 0$. This is false for $x = 0$ because $0^2 = 0 
gtr 0$. Thus, $ho$ is not reflexive.(ii) Symmetric: If $x ho y$, then $xy > 0$. Since multiplication is commutative, $yx > 0$, which implies $y ho x$. Thus, $ho$ is symmetric.(iii) Transitive: Suppose $x ho y$ and $y ho z$. Then $xy > 0$ and $yz > 0$. Since $y 
eq 0$ (as $xy > 0$), we have $y^2 > 0$. Multiplying the inequalities, we get $(xy)(yz) > 0$, which simplifies to $y^2(xz) > 0$. Since $y^2 > 0$, we must have $xz > 0$. Thus, $x ho z$. Therefore, $ho$ is transitive.Conclusion: The relation is symmetric and transitive.

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

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