Q. Let a relation $R$ on the set $N$ of natural numbers be defined as $(x,\ y) \Longleftrightarrow x^{2} - 4xy + 3y^{2} = 0\forall\ x,\ y \in N$. The relation $R$ is
A) Reflexive
B) Symmetric
C) Transitive
D) An equivalence relation
Explanation
(a)
Given, $(x,\ y) \Leftrightarrow x^{2} - 4xy + 3y^{2} = 0$
Or $(x,\ y) \Leftrightarrow (x - y)(x - 3y) = 0$
(i) Reflexive
$\ \ \ \ \ \ \ \ \ \ xRx \Rightarrow (x - x)(x - 3x) = 0$
$\therefore$ It is reflexive.
(ii) Symmetric
Now, $xRy \Leftrightarrow (x - y)(x - 3y) = 0$
And, $yRx \Leftrightarrow (y - x)(y - 3x) = 0 \Rightarrow xRy \neq yRx$
$\therefore$ It is not symmetric.
Similarly, it is not transitive.
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.