Let a relation R on the Set N of natural numbers be defined as (x,y) ~~ ~~belongs to R if and only if x^{2}-4xy+3y^{2}=0 for all x,y belongs to N.Verify that R is reflexive but not symmetric and transiitve.

Let x=y,

${x}^{2}-4xy+3{y}^{2}={x}^{2}-4{x}^{2}+3{x}^{2}=0$

So, it is reflexive.

Symmetric:

Interchange x and y

${y}^{2}-4xy+3{x}^{2}=0\phantom{\rule{0ex}{0ex}}3{x}^{2}-4xy+{y}^{2}=0$

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

${x}^{2}-4xy+3{y}^{2}=0...\left(1\right)\phantom{\rule{0ex}{0ex}}{y}^{2}-4yz+3{z}^{2}=0...\left(2\right)\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}Byu\mathrm{sin}gthesetwowecannotprove{x}^{2}-4xz+3{y}^{2}=0.\phantom{\rule{0ex}{0ex}}Sowecannotsay(x,z)\in R.\phantom{\rule{0ex}{0ex}}$

So, R is not transitive.

**
**