JEE Main20203 Sep 2020Evening ShiftMathematicsSets and RelationsActual
Let R 1 and R 2 be two relations defined as follows : R 1 = a , b ∈ R 2 : a 2 + b 2 ∈ Q and R 2 = a , b ∈ R 2 : a 2 + b 2 ∉ Q , where Q is the set of all rational numbers, then
Options
- AR 1 is transitive but R 2 is not transitive.
- BR 2 is transitive but R 1 is not transitive.
- CNeither R 1 nor R 2 is transitive.
- DR 1 and R 2 are both transitive.
Correct answer
C. Neither R 1 nor R 2 is transitive.
Step-by-step solution
For R 1 let a = 1 + 2 ,   b = 1 - 2 ,   c = 8 1 4 a R 1 b ⇒ a 2 + b 2 = 1 + 2 2 + 1 - 2 2 = 6 ∈ Q b R 1 c ⇒ b 2 + c 2 = 1 - 2 2 + 8 1 4 2 = 3 ∈ Q a R 1 c ⇒ a 2 + c 2 = 1 + 2 2 + 8 1 4 2 = 3 + 4 2 ∉ Q ∴ R 1 is not transitive. For R 2 let a = 1 + 2 ,   b = 2 ,   c = 1 - 2 a R 2 b ⇒ a 2 + b 2 = 1 + 2 2 + 2 2 = 5 + 2 2 ∉ Q b R 2 c ⇒ b 2 + c 2 = 2 2 + 1 - 2 2 = 5 - 2 2 ∉ Q a R 2 c ⇒ a 2 + c 2 = 1 + 2 2 + 1 - 2 2 = 6 ∈ Q