Question

Let a relation $R$ in the set $N$ of natural numbers be defined by $(x, y)\Leftrightarrow x^2 - 4xy + 3y^2 = 0\,\forall\,x, y \in\,N.$The relation $R$ is

• A. reflexive
• B. symmetric
• C. transitive
• D. an equivalence relation

Answer 1

Answer: (A)

reflexive

Answer 2

The correct answer is A:reflexive
Given that;
R is a relation on N defined by;
$xRy=x^2-4xy+3y^2=0$
⇒$(x-y)(x-3y)=0-(i)$
The given equation is reflexive if x=a and y=a
i.e., $(a,a)\in R\forall N$ and symmetric
$(1,3)$ satisfies the equation (i)
$(1,3)\in R$
$(1-3)(1-3\times3)$
$=16\neq 0$
$\therefore (1,3)\not\in R$
$\therefore$ Not symmetric
Let us assume the function is transitive for (9,1)
Let's Substitute and test $\therefore (9-1)(9-3\times1)$
$=48\neq0$
Hence, not transitive $\therefore (9,1)\not\in R$