Question

The relation $R$ defined on the set $N$ of natural number 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 the above

Answer 1

Answer: (C)

Not symmetric but reflexive

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 $xϵN$
$\therefore 2x^2-3x\cdot x+x^2=2x^2-3x^2+x^2=0$
$\therefore xRx$.
$\therefore R$ is Reflexive.
(ii) Symmetric : Let $x,yϵN$, such that $(x,y)\in R$
$\Rightarrow 2x^2-3xy+y^2=0$

But, this does not implies $2y^2-3xy+x^2=0$
$\Rightarrow (y,x)\notin R$
$(x,y)ϵR$ but $(y,x)\notin R$
$\therefore R$ is not symmetric.