According to the question ,

Given set S = {…….,-2,-1,0,1,2 …..}

And R = {(a, b) : a,b ∈ S and |a – b| ≤ 1 }

Formula

For a relation R in set A

Reflexive

The relation is reflexive if (a , a) ∈ R for every a ∈ A

Symmetric

The relation is Symmetric if (a , b) ∈ R , then (b , a) ∈ R

Transitive

Relation is Transitive if (a , b) ∈ R & (b , c) ∈ R , then (a , c) ∈ R

Equivalence

If the relation is reflexive , symmetric and transitive , it is an equivalence relation.

Check for reflexive

Consider , (a,a)

∴ |a – a| ≤ 1 and which is always true.

Ex_if a=2

∴ |2-2| ≤ 1 ⇒ 0 ≤ 1 which is true.

Therefore , R is reflexive ……. (1)

Check for symmetric

a R b ⇒ |a – b| ≤ 1

b R a ⇒ |b – a| ≤ 1

Both can be true.

Ex _ If a=2 and b=1

∴ |2 – 1| ≤ 1 is true and |1–2| ≤ 1 which is also true.

Therefore , R is symmetric ……. (2)

Check for transitive

a R b ⇒ |a – b| ≤ 1

b R c ⇒ |b – c| ≤ 1

∴|a – c| ≤ 1 will not always be true

Ex _a=-5 , b= -6 and c= -7

∴ |6-5| ≤ 1 , |7 – 6| ≤ 1 are true But |7 – 5| ≤ 1 is false.

Therefore , R is not transitive ……. (3)

Now , according to the equations (1) , (2) , (3)

Correct option will be (A)