Question

The relation $R$ on the set of natural numbers $N$ is defined as $xRy⟺x^2-4xy+3y^2=0,x,y\in N$ then $R$ is

• A. reflexive but neither symmetric nor transitive relation.
• B. symmetric but neither reflexive nor transitive relation
• C. transitive but neither reflexive nor symmetric relation
• D. an equivalence relation

Answer 1

The correct option is A reflexive but neither symmetric nor transitive relation.

$xRy⟺x^2-4xy+3y^2=0$
$x^2-xy-3xy+3y^2=0$
$x(x-y)-3y(x-y)=0$
$(x-3y)(x-y)=0$
$\therefore (x,y)\in R$ iff $(x-3y)(x-y)=0$

As $(x-3x)(x-x)=0$ $\forall x\in N$
$\Rightarrow (x,x)\in R$
so $R$ is a reflexive relation

It can be observed that $(3,1)\in R$ but $(1,3)\notin R$ as $(1-9)(1-3)\ne 0$
so $R$ is not a symmetric relation

As $(3,1)$ and $(1,\frac{1}{3})\in R$ but $(3,\frac{1}{3})\notin R$
so $R$ is not a transitive relation.