Let a relation R on the set - of natural numbers be defined as x- y - x2 -4xy-3y2-0 - x- y - Then the relation R is

Given (x, y) nbsp;⇔ x^2 –4xy+3y^2=0 nbsp; or (x, y) nbsp;⇔ (x y)(x 3y)=0 or nbsp;(x, y) nbsp;⇔ x=y or x=3y nbsp; (1) Reflexive : Clearly, (x,x) ∈ R ...

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Byju's Answer

Standard XIII

Mathematics

Reflexive Relations

Let a relatio...

Question

Let a relation $R$ on the set $N$ of natural numbers be defined as $(x, y) \Leftrightarrow x^{2} - 4 x y + 3 y^{2} = 0 \forall x, y \in N .$ Then the relation $R$ is

reflexive

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symmetric

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transitive

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an equivalence relation

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Solution

The correct option is A reflexive
Given $(x, y) \Leftrightarrow x^{2} - 4 x y + 3 y^{2} = 0$
or $(x, y) \Leftrightarrow (x - y) (x - 3 y) = 0$
or $(x, y) \Leftrightarrow x = y$ or $x = 3 y$

$(1)$ Reflexive :
Clearly, $(x, x) \in R$ for all $x \in N$ as $x = y$
$\Rightarrow R$ is reflexive.

$(2)$ Symmetric :
$(3, 1) \in R$ but $(1, 3) \notin R$
$\Rightarrow R$ is not symmetric.

$(2)$ Transitive :
$(9, 3), (3, 1) \in R$ but $(9, 1) \notin R$
So, $R$ is not transitive.

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Let a relation R on the set N of natural numbers be defined as (x,y)⇔x2−4xy+3y2=0 ∀ x,y∈N. Then the relation R is

Let a relation $R$ on the set $N$ of natural numbers be defined as $(x, y) \Leftrightarrow x^{2} - 4 x y + 3 y^{2} = 0 \forall x, y \in N .$ Then the relation $R$ is