Question

Let a relation R on the Set N of natural numbers be defined as (x,y) belongs to R if and only if x²-4xy+3y²=0 for all x,y belongs to N.Verify that R is reflexive but not symmetric and transiitve.

Answer 1

Reflexive:
Let x=y,
$x^2-4xy+3y^2=x^2-4x^2+3x^2=0$
So, it is reflexive.

Symmetric:
Interchange x and y
$$\begin{gathered} y^2-4xy+3x^2=0 \\ 3x^2-4xy+y^2=0 \end{gathered}$$
This is not same as given equation. So it is not symmetric.

Transitive:
Let (x,y) and (y,z) satisfies the given relation.
Then we get
$$\begin{gathered} x^2-4xy+3y^2=0...\left(1\right) \\ {y}^2-4yz+3z^2=0...\left(2\right) \\ Byu\sin gthesetwowecannotprove{x}^2-4xz+3y^2=0. \\ Sowecannotsay(x,z)\in R. \end{gathered}$$
So, R is not transitive.