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
Answer & explanation
Correct answer: option 4
|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