Let a relation in the set of natural numbers be defined as (, ) ⇔ 2 − 4 + 3
2 =0∀ , ∈ . The relation is
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We have R={(x,y):x
2
−4xy+3y
2
=0,x,y∈N}.
Let x∈N.x
2
−4xx+3x
2
=4x
2
−4x
2
=0
∴(x,x)∈R.
∴R is reflexive
We have (3)
2
−4(3)(1)+3(1)
2
=9−12+3=0
∴(3,1)∈R.
Also (1)
2
−4(1)(3)+3(3)
2
=1−12+27=16
=0
∴(1,3)∈
/
R.
∴R is not symmetric.
(9,3)∈R because
(9)
2
−4(9)(3)+3(3)
2
=81−108+27=0
Also (3,1)∈R because
(3)
2
−4(3)(1)+3(1)
2
=9−12+3=0
Now, (9,1)∈R if (9)
2
−4(9)(1)+3(1)
2
=0
if 81−36+3=48
=0,
which is not so.
∴(9,3),(3,1)∈R and (9,1)∈
/
R
∴R is not transitive.
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