Let a relation R on the set N of natural number be defined as (x,y)∈R If and only If x2−4xy+3y2=0 for all,x,y∈N and the relation is

we have, $R = {(x, y); x^{2} - 4 xy + 3 y^{2} = 0, x, y \in N$

Let $x \in N, x^{2} - 4 x \cdot x + 3 x^{2} = 0$

$∴ (x, x) \in R$

R is reflexive

we have,

$\begin{aligned} (3)^{2} - 4 (3) (1) + 3 (1)^{2} = 9 - 12 + 3 = 0 \\ (3, 1) \in R \end{aligned}$

Also, $(1)^{2} - 4 (1) (3) + 3 (3)^{2} = 1 - 12 + 27 = 16 \neq 0$

$(3)^{2} - 4 (3) (1) + 3 (1)^{2} = 0$

now, $(9, 1) \in R if (9)^{2} - 4 (9) (1) + 3 (1)^{2} = 0$

i.e $48 \neq 0$

which is not so $(9, 3), (3, 1) \in R and (9, 1) \notin R$

$∴$ R is not transitive.

Let a relation R on the set N of natural number be defined as $(x, y) \in R$ If and only If $x^{2} - 4 xy + 3 y^{2} = 0$ for all, $x, y \in N$ and the relation is