Question

$x^2=xy$ is a relation which is

• A. Symmetric
• B. Reflexive
• C. Transitive
• D. All of these

Answer 1

The correct option is D

All of these

Explanation for the correct option:

Given that, $x^2=xy$

The Reflexive property states that for every real number $x$; $x=x$

$\Rightarrow xRx:x^2=x·x\Rightarrow x=x$.

Thus the given relation is reflexive.

The Symmetric property states that for all real numbers $x,y$ if $x=y$ the $y=x$

$$\begin{gathered} \Rightarrow xRy:x^2=x·y\Rightarrow x=y \\ yRx:y^2=x·y\Rightarrow y=x \end{gathered}$$

Thus the given relation is symmetric

The Transitive property states that for all real numbers $x,y,z$; if $x=y$ and $y=z$ then $x=z$

$$\begin{gathered} \Rightarrow xRy:x^2=xy\Rightarrow x=y \\ yRz:y^2=yz\Rightarrow y=z \\ xRz:x^2=xz\Rightarrow x=z \end{gathered}$$

Thus the given relation is transitive

Hence, the correct option is option $\left(D\right)$ i.e. All of these