Question

81. Which of the following is not correct for relation \( R \) on the set of real numbers? [JEE(Main)-2021] (1) \( (x, y) \in R \Leftrightarrow|x-y| \leq 1 \) is reflexive and symmetric. (2) \( (x, y) \in R \Leftrightarrow 0|x|-|y| \leq 1 \) is neither transitive nor symmetric (3) \( (x, y) \in R \Leftrightarrow 0<|x-y| \leq 1 \) is symmetric and transitive (4) \( (x, y) \in R \Leftrightarrow|x|-|y| \leq 1 \) is reflexive but not symmetric

Answer 1

Answer: (x, y) ∈ R \(\Leftrightarrow\) 0 < |x – y|\(\leq\) 1 is symmetric and transitive.

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.