Question

The relation $R$ given by $\left\{\left(x , y\right) : x^{2} - 3 x y + 2 y^{2} = 0 , \forall x , y \in R\right\}$ is

• A. reflexive but not symmetric
• B. symmetric but not transitive
• C. symmetric and transitive
• D. an equivalence relation

Answer 1

Answer: (A)

$\because x^{2}-3xy+2y^{2}=0\Rightarrow x^{2}-xy-2xy+2y^{2}=0$
$\Rightarrow x\left(x - y\right)-2y\left(x - y\right)=0$
$\Rightarrow \left(x - 2 y\right)\left(x - y\right)=0$
$\Rightarrow x=y$ or $x=2y$
Now, as in $R$ all ordered pairs $\left(x , x\right)$ are present, therefore it is reflexive.
Now, $\left(4 , 2\right)\in R$ as $4=2\left(2\right)$
but $\left(2,4\right)\notin R$ as $2\neq 2\left(4\right)$
$\therefore $ It is not symmetric
Also $\left(4,2\right)$ & $\left(2,1\right)\in R$ but $\left(4,1\right)\notin R$
$\therefore $ It is not transitive