Question

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

• A. symmetric
• B. reflexive
• C. transitive
• D. none of these

Answer 1

Answer: (C)

$xRx\Rightarrow x^2=x\cdot x$ Hence reflexive
$xRy\Rightarrow x^2=xy.$
$\Rightarrow x=yyRx$
$\Rightarrow y^2=xy$
$\Rightarrow y=x$ Hence symmetric
$xRy$ and $yRz$
$\Rightarrow x^2=xy$
$\Rightarrow x=y.$ Also $y^2=yz$
$\Rightarrow y=z$
Since $x=z,x\cdot x=z\cdot x$
$\Rightarrow x^2=xz$
$\Rightarrow xRz$. Hence transitive.