# 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.

Reflexive:
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

So, R is not transitive.

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