Question

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

• A. symmetric
• B. reflexive and transitive
• C. transitive
• D. None of the above

Answer 1

The correct option is B

reflexive and transitive

Reflexive relation: A relation $R$ is reflexive on a set $A$, if $\left(a,a\right)\in R$ , $\forall a\in A$

Symmetric relation: A relation $R$ is symmetric on a set $A$, if $\left(a,b\right)\in R$ then $\left(b,a\right)\in R$ , $\forall a,b\in A$

Transitive relation: A relation $R$ is reflexive on a set $A$, if $\left(a,b\right)\in R$ and $\left(b,c\right)\in R$ then $\left(a,c\right)\in R$, $\forall a,b,c\in A$

Explanation for the correct option:

$B)$ reflexive and transitive

Let $R$ be the relation

Reflexive: $xRx$

$x^2=x·x$

Hence, the relation is reflexive.

Symmetric:

If $xRy$ , then

$x^2=xy$

If $yRx$, then

$y^2=yx$

$\Rightarrow$ $y^2=xy$

Hence, the relation is not symmetric.

Transitive:

If $xRy$ , then $x^2=xy$ ….(1)

If $yRz$ , then $y^2=yz$ ….(2)

Multiply equations (1) and (2)

$x^2\times y^2=xy\times yz$

$\Rightarrow$$x^2\times y^2=xy\times yz$

$\Rightarrow$ $x^2{y}^2=x{y}^{2}z$

$\Rightarrow$ $x^2=xz$

Since $x^2=xz$ so $xRz$.

Hence, the relation is transitive.

Therefore, the correct option is $B)$ reflexive and transitive.