Which of the following is not correct for relation R

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

$(x, y) \in R \Leftrightarrow 0<|x|-|y| \leq 1$ is neither transitive nor symmetric.
$(x, y) \in R \Leftrightarrow 0<|x-y| \leq 1$ is symmetric and transitive.
$(x, y) \in R \Leftrightarrow|x|-|y| \leq 1$ is reflexive but not symmetric.
$(x, y) \in R \Leftrightarrow|x-y| \leq 1$ is reflexive and symmetric.

Correct Option: , 2

Solution:

Note that $(1,2)$ and $(2,3)$ satisfy $0<|x-y| \leq 1$

but $(1,3)$ does not satisfy it so

$0 \leq|x-y| \leq 1$ is symmetric but not transitive

So, (2) is correct.