Each of the following defines a relation on N:

x y is square of an integer x, y ∈N

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

Let R = {(x,y): x y is square of an integer, ∀ x,y ∈ N } be a relation defined on N.

R is reflexive ∵ x² is square of an integer ∀ x ∈ N ⇒ (x,x) ∈ R

Let (x,y) ∈ R ∀ x, y ∈ N

⇒ x y is square of an integer

⇒ y x is square of an integer

⇒ (y,x) ∈ R

⇒ R is symmetric

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

⇒ x y is square of an integer and yz is square of an integer

let xy = p² and yz = q² for some p,q ∈ Z

⇒

⇒ , which is square of an integer

⇒ (x,z) ∈ R

⇒ R is transitive.

Thus, R is reflexive, symmetric and transitive.