Let A be set of points on the plane.

Let R = {(P, Q) : OP = OQ} be a relation on A where O is the origin.

To prove R is an equivalence relation, we need to show that R is reflexive, symmetric and transitive on A.

Proof :

To prove that relation is equivalence, we need to prove that it is reflexive, symmetric and transitive.

Reflexivity : For Reflexivity, we need to prove that-

(a, a) ∈ R

Let p ∈ A

Since OP = OP ⇒ (P, P) ∈ R

⇒ R is reflexive

Symmetric : For Symmetric, we need to prove that-

If (a, b) ∈ R, then (b, a) ∈ R

Let (P, Q) ∈ R for P, Q ∈ R

Then OP = OQ

⇒ Op = OP

⇒ (Q, P) ∈ R

⇒ R is symmetric

Transitive: For Transitivity, we need to prove that-

If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

Let (P, Q) ∈ R and (Q, S) ∈ R

⇒ OP = OQ and OQ = OS

⇒ OP = OS

⇒ (P, S) ∈ R

⇒ R is transitive

Thus, R is an equivalence relation on A