Which of the following is not correct for relation R on the set of real numbers?

Which of the following is not correct for relation R on the set of real numbers?
  1. (x,y)∈R⇔|x|-|y|≤1 is reflexive but not symmetric.
  2. (x,y)∈R⇔|x-y|≤1 is reflexive and symmetric.
  3. (x,y)∈R⇔0<|x-y|≤1 is symmetric and transitive.
  4. (x,y)∈R⇔0<|x|-|y|≤1 is not transitive but symmetric.

Solution

We know that (x, y)∈R , (y, x)∈R ⇒symmetric

(x, y)∈R , (y, z)∈R & (z, x)∈R⇒transitive
0<|x-y|≤10<|y-x|≤1is symmetric

let 

0<|x-y|≤1                                         .......(1)
(1,2)∈R and (2,3)∈R satisfy the equation (1)
but (1,3)∉R not satisfied
Hence 0<|x-y|≤1 is symmetric but not transitive relation

Asked in: JEE Main 2021 (31 Aug Shift 1)

Practice more Sets and Relations questions on Aicharya