Correct Answer is (C) transitive but neither reflexive nor symmetric

Given set S = {…….,-2,-1,0,1,2 …..}

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

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 and which is not always true.

Ex_if a=-2

∴ |-2| ≤ -2 ⇒ 2 ≤ -2 which is false

Therefore , R is not reflexive ……. (1)

Check for symmetric

a R b ⇒ |a| ≤ b

b R a ⇒ |b| ≤ a

Both cannot be true.

Ex _ If a=-2 and b=-1

∴ 2 ≤ -1 is false and 1 ≤ -2 which is also false

Therefore , R is not symmetric ……. (2)

Check for transitive

a R b ⇒ |a| ≤ b

a R b ⇒ |a| ≤ b

∴ |a| ≤ c

Ex _a=-5 , b= 7 and c=9

∴ 5 ≤ 7 , 7 ≤ 9 and hence 5 ≤ 9

Therefore , R is transitive ……. (3)

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

Correct option will be (C)