Correct Answer is (C) . reflexive and transitive but not symmetric

Given set Z = {1, 2, 3 ,4 …..}

And R = {(a, b) : a,b ∈ Z 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) (b,b)

∴ a ≥ a and b ≥ b which is always true.

Therefore , R is 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 true but 1 ≥ 2 which is false.

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

Check for transitive

a R b ⇒ a ≥ b

b R c ⇒ b ≥ c

∴ a ≥ c

Ex _a=5 , b=4 and c=2

∴ 5≥4 , 4≥2 and hence 5≥2

Therefore , R is transitive ……. (3)

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

Correct option will be (C)