Jee mains pyq 2022 physics

Question

Let R1 and R2 be two relations defined on R  by αR1 b ⇔ ab ≥ 0 and α R2b ⇔ a ≥ b , then (A) R1 is an equivalence relation but not R2 (B) R2 is an equivalence relation but not R1 (C) both R1 and R2 are equivalence relations (D) neither R1 nor R2 is an equivalence relation

Answer 1

Correct option: (D) neither R₁ nor R₂ is an equivalence relation.

R 1 = { xy ≥ 0 , x , y ∈ R } 

For reflexive x × x ≥ 0 which is true. 

For symmetric If xy ≥ 0 ⇒ yx ≥ 0 

If x = 2, y = 0 and z = -2 

Then x . y ≥ 0 & y . z ≥ 0 but x.z ≥ 0 is not true 

⇒ not transitive relation. 

⇒ R₁ is not equivalence 

R₂ if a ≥ b it does not implies b ≥ a 

⇒ R₂ is not equivalence relation