If R is a relation on Z (set of all integers) defined by xRy, iff |x – y| ≤ 1, then

(a) R is reflexive
(b) R is symmetric
(c) R is transitive
(d) R is not symmetric
(e) R is not transitive

Choose the most appropriate answer from the options given below

(a), (b) and (e) only

|x - y| ≤ 1   →   Relation on Z

for every x ∈ Z  ⇒  |x - x| = 0 ≤ 1  ⇒  (x, x) ∈ R

reflexive

for (x, y) ∈ R  ⇒  |x - y| ≤ 1  ⇒  |y - x| ≤ 1  ⇒  (y, x) ∈ R

symmetric

for (x, y) ∈ R  , (y, z) ∈ R  ⇒  |x - y| ≤ 1      |y - z| ≤ 1

eg: (2, 1)∈ R        (2, 3) ∈ R

⇒ |z - 1| ≤ 1     |2 - 3| ≤ 1    but  |(1 - 3)| ≤ 1  (false)

So it is not transitive

Option: 4