Assume that R is a relation on the set Z of integers and it is given by \((x, y) \in R \Leftrightarrow |x - y| \le 1\). Then, R is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Reflexive and symmetric but not transitive
Reflexive: for every integer x,
\(|x - x| = 0 \le 1\)
so \((x, x) \in R\).
Symmetric: \(|y - x| = |x - y|\), so if \(|x - y| \le 1\) then \(|y - x| \le 1\), and \((x, y) \in R\) gives \((y, x) \in R\).
Not transitive: a counter-example is x = 1, y = 2, z = 3:
\(|1 - 2| = 1 \le 1\), so \((1, 2) \in R\)
\(|2 - 3| = 1 \le 1\), so \((2, 3) \in R\)
\(|1 - 3| = 2 \gt 1\), so \((1, 3) \notin R\)
So R is reflexive and symmetric but not transitive.