Let a relation R on the set N of natural numbers be defined as ( x , y ) Leftrightarrow x 2-4 xy +3 y 2=0 ∀ x , y ∈ N. The relation R is

Question

Let a relation R on the set N of natural numbers be defined as ( x , y ) Leftrightarrow x 2-4 xy +3 y 2=0 ∀ x , y ∈ N. The relation R is (A) Reflexive (B) Symmetric (C) Transitive (D) An equivalence relation

Tardigrade Admin · Accepted Answer

Given, (x, y) Leftrightarrow x2-4 x y+3 y2=0 Or (x, y) Leftrightarrow(x-y)(x-3 y)=0 (i) Reflexive x R x ⇒(x-x)(x-3 x)=0 ∴ It is reflexive. (ii) Symmetric Now, x R y Leftrightarrow(x-y)(x-3 y)=0 And, y R x Leftrightarrow(y-x)(y-3 x)=0 ⇒ x R y ≠ y R x ∴ It is not symmetric. Similarly, it is not transitive.

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

Reflexive

Symmetric

Transitive

An equivalence relation

Given, (x,y)⇔x2−4xy+3y2=0 Or (x,y)⇔(x−y)(x−3y)=0 (i) Reflexive xRx⇒(x−x)(x−3x)=0 ∴ It is reflexive. (ii) Symmetric Now, xRy⇔(x−y)(x−3y)=0 And, yRx⇔(y−x)(y−3x)=0 ⇒xRy=yRx ∴ It is not symmetric. Similarly, it is not transitive.

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