Question

The relation $R$ defined on the set $N$ of natural numbers by $xRy\iff 2x^2-3xy+y^2=0$ is

• A. symmetric but not reflexive
• B. only symmetric
• C. not symmetric but reflexive
• D. none of these

Answer 1

Answer: (C)

The relation $R$ on the set of natural number $N$ is defined by
$xRy\iff 2x^2-3xy+y^2=0.\forall x,y\in N$
(i) Reflexive : Let $x\in N$
$\therefore 2x^2-3x\cdot x+x^2=2x^2-3x^2+x^2=0$
$\therefore xRx$
$\therefore R$ is Reflexive.
(ii) Symmetric : Let $x,y\in N$, such that $(x,y)\in R$
$\therefore (x,y)\in R\Rightarrow 2x^2-3xy+y^2=0$
$\Rightarrow 2y^2-3xy+x^2=0\forall x\ne y$
$\Rightarrow (y,x)\notin R$
$(x,y)\in R$ but $(y,x)\notin R$
$\therefore R$ is not symmetric.