Each of the following defines a relation on N:

x is greater than y, x, y ∈N

Determine which of the above relations are reflexive, symmetric and transitive.

Let R = {(x,y): x is greater than y ∀ x,y ∈ N } be a relation defined on N.

Now,

We observe that, any element x ∈ N cannot be greater than itself.

⇒ (x,x) ∉ R ∀ x ∈ N

⇒ R is not reflexive.

Let (x,y) ∈ R ∀ x, y ∈ N

⇒ x is greater than y

But y cannot be greater than x if x is greater than y.

⇒ (y,x) ∉ R

For e.g., we observe that (5,2) ∈ R i.e 5 > 2 but 2 ≯ 5 ⇒ (2,5) ∉ R

⇒ R is not symmetric

Let (x,y) ∈ R and (y,z) ∈ R ∀ x, y,z ∈ N

⇒ x > y and y > z

⇒ x > z

⇒ (x,z) ∈ R

For e.g., we observe that

(5,4) ∈ R ⇒ 5 > 4 and (4,3) ∈ R ⇒ 4 > 3

And we know that 5 > 3 ∴ (5,3) ∈ R

⇒ R is transitive.

Thus, R is transitive but not reflexive not symmetric.