Let N be the set of natural numbers and a relation R on N be defined by R = x , y ∈ N × N : x 3 -…

Let N be the set of natural numbers and a relation R on N be defined by R=x,y∈N×N:x3-3x2y-xy2+3y3=0. Then the relation R is
  1. symmetric but neither reflexive nor transitive
  2. reflexive but neither symmetric nor transitive
  3. reflexive and symmetric, but not transitive
  4. an equivalence relation

Solution

x3-3x2y-xy2+3y3=0

⇒xx2-y2-3yx2-y2=0

⇒x-3yx-yx+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.

Asked in: JEE Main 2021 (27 Jul Shift 2)

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