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Q. The relation $R$ defined on the set $N$ of natural numbers by $ x\, R\, y\Leftrightarrow\,2x^2-3xy+y^2=0$ is

UPSEEUPSEE 2010Relations and Functions - Part 2

Solution:

The relation $R$ on the set of natural number $N$ is defined by
$x R y \Leftrightarrow 2 x^{2}-3 x y+y^{2}=0 . \forall x, y \in N$
(i) Reflexive : Let $x \in N$
$\therefore 2 x^{2}-3 x \cdot x+x^{2}=2 x^{2}-3 x^{2}+x^{2}=0$
$\therefore x R x$
$\therefore R$ is Reflexive.
(ii) Symmetric : Let $x, y \in N$, such that $(x, y) \in R$
$\therefore (x, y) \in R \Rightarrow 2 x^{2}-3 x y+y^{2}=0$
$\Rightarrow 2 y^{2}-3 x y+x^{2}=0 \,\,\,\forall x \neq y$
$\Rightarrow (y, x) \notin R$
$(x, y) \in R$ but $(y, x) \notin R$
$\therefore R$ is not symmetric.