Question

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

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

Answer 1

Answer: (A)

$\because x^2-3xy+2y^2=0\Rightarrow x^2-xy-2xy+2y^2=0$
$\Rightarrow x\left(x-y\right)-2y\left(x-y\right)=0$
$\Rightarrow \left(x-2y\right)\left(x-y\right)=0$
$\Rightarrow x=y$ 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 $2\ne 2\left(4\right)$
$\therefore$ 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