Let R 1 and R 2 be two relations defined on ℝ by a R 1 b ⇔ a b ≥ 0 and a R 2 b ⇔ a ≥ b , then

Step-by-step solution

Given, R 1 = a b ≥ 0 , a , b ∈ R For reflexive a × a ≥ 0 which is true, For symmetric If a b ≥ 0 ⇒ b a ≥ 0 which is true, If a = 2 , b = 0 and c = - 2 Then a · b ≥ 0 and b · c ≥ 0 but a · c ≥ 0 is not true ⇒ not transitive relation ⇒   R 1 is not equivalence For R 2 if a ≥ b it does not implies b ≥ a ⇒   R 2 is not equivalence relation So, neither R 1 nor R 2 is an equivalence relation