Question

Let a relation \( R \) on the set of natural numbers be defined by \( (x, y) \in R \) if and only if \( x^2 - 4xy + 3y^2 = 0 \) for all \( x, y \in \mathbb{N} \). Then the relation is:

• A. reflexive
• B. symmetric
• C. transitive
• D. reflexive and symmetric but not transitive
• E. an equivalence relation

Hint:

A relation is reflexive if for every \( x \in \mathbb{N} \), \( (x, x) \in R \).

Answer 1

The Correct Option is A

To check if the relation is reflexive, we check if \( x = y \) satisfies the equation. For \( x = y \), the equation becomes: \[ x^2 - 4x^2 + 3x^2 = 0 \] This simplifies to: \[ 0 = 0 \] Hence, the relation is reflexive. However, we do not need to check for symmetry and transitivity in this case as the correct answer is already found to be reflexive.