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JEE Main202127 Jul 2021Evening ShiftMathematicsSets and RelationsActual

Let N be the set of natural numbers and a relation R on N be defined by R = x , y ∈ N × N : x 3 - 3 x 2 y - x y 2 + 3 y 3 = 0 . Then the relation R is

Options

  1. Asymmetric but neither reflexive nor transitive
  2. Breflexive but neither symmetric nor transitive
  3. Creflexive and symmetric, but not transitive
  4. Dan equivalence relation

Correct answer

B. reflexive but neither symmetric nor transitive

Step-by-step solution

x 3 - 3 x 2 y - x y 2 + 3 y 3 = 0 ⇒ x x 2 - y 2 - 3 y x 2 - y 2 = 0 ⇒ x - 3 y x - y x + y = 0 Now, x = y   ∀ x , y ∈ N × N so its a reflexive relation. But not symmetric & transitive Since, 3 , 1 satisfies but 1 , 3 does not. Also 3 , 1 & 1 , - 1 satisfies but 3 , - 1 does not.

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