Let R 1 and R 2 be two relations defined as follows : R 1 = a ,   b ∈ R 2 : a 2 + b 2 ∈ Q…

Let R1 and R2 be two relations defined as follows :

R1=a, b∈R2:a2+b2∈Q and R2=a, b∈R2:a2+b2∉Q, where Q is the set of all rational numbers, then

  1. R1 is transitive but R2 is not transitive.
  2. R2  is transitive but R1 is not transitive.
  3. Neither R1 nor R2 is transitive.
  4. R1 and R2 are both transitive.

Solution

For R1 let a=1+2, b=1-2, c=814

aR1b⇒a2+b2=1+22+1-22=6∈Q

bR1c⇒b2+c2=1-22+8142=3∈Q

aR1c⇒a2+c2=1+22+8142=3+42∉Q

∴R1 is not transitive.

For R2 let a=1+2, b=2, c=1-2

aR2b⇒a2+b2=1+22+22=5+22∉Q

bR2c⇒b2+c2=22+1-22=5-22∉Q

aR2c⇒a2+c2=1+22+1-22=6∈Q

∴R2 is not transitive.

Asked in: JEE Main 2020 (03 Sep Shift 2)

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