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,
x2-4xy+3y2=x2-4x2+3x2=0
So, it is reflexive.

Symmetric:
Interchange x and y
y2-4xy+3x2=03x2-4xy+y2=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
x2-4xy+3y2=0 ...(1)y2-4yz+3z2=0 ...(2)By using these two we cannot prove x2-4xz+3y2=0.So we cannot say (x,z) R.
So, R is not transitive.

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