Question

Let R 1 and R 2 be two relations defined on ℝ by a R 1 b ⇔ ab ≥ 0 and a R 2 b ⇔ a ≥ _b. _Then,

• A. R 1 is an equivalence relation but not R 2
• B. R 2 is an equivalence relation but not R 1
• C. Both R 1 and R 2 are equivalence relations
• D. Neither R 1 nor R 2 is an equivalence relation

Answer 1

Answer: (D)

Neither R 1 nor R 2 is an equivalence relation

Answer 2

_The correct answer is (D):
aR_1 b ⇔ ab ≥ 0
So, definitely (a , a) ∈ R 1 as a 2 ≥ 0
If (a , b) ∈ R 1 ⇒ (b , a) ∈ R 1
But if (a , b) ∈ R 1, (b , c) ∈ R 1
⇒ Then (a , c) may or may not belong to R 1
{Consider a = -5, b = 0, c = 5 so (a , b) and (b , c) ∈ R 1 but ac < 0}
So, R 1 is not equivalence relation
a R 2 b ⇔ a ≥ _b
_(a , a) ∈ R 2 ⇒ so reflexive relation
If (a , b) ∈ R 2 then (b , a) may or may not belong to R 2
⇒ So not symmetric
Hence it is not equivalence relation