$x^{2} = x y$ is a relation which is:

Symmetric

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Reflexive

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Transitive

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All of these

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Solution

The correct option is D All of these
$x R x \Rightarrow x^{2} = x . x$ Hence reflexive

$x R y \Rightarrow x^{2} = x y . \Rightarrow x = y$ $y R x \Rightarrow y^{2} = x y \Rightarrow y = x$ Hence symmetric

$x R y$ and $y R z$ $\Rightarrow x^{2} = x y \Rightarrow x = y .$ Also $y^{2} = y z \Rightarrow y = z$

Since $x = z, x . x = z . x \Rightarrow x^{2} = x z \Rightarrow x R z$ . Hence transitive.

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