In order to show R is an equivalence relation, we need to show R is Reflexive, Symmetric and Transitive.

Given that, A be the set of all points in a plane and O be the origin. Then, R = {(P, Q) : P, Q ∈ A and OP = OQ)}

Now,

R is Reflexive if (P,P) ∈ R ∀ P ∈ A

∀ P ∈ A , we have

OP=OP

⇒ (P,P) ∈ R

Thus, R is reflexive.

R is Symmetric if (P,Q) ∈ R ⇒ (Q,P) ∈ R ∀ P, Q ∈ A

Let P, Q ∈ A such that,

(P,Q) ∈ R

⇒ OP = OQ

⇒ OQ = OP

⇒ (Q,P) ∈ R

Thus, R is symmetric.

R is Transitive if (P,Q) ∈ R and (Q,S) ∈ R ⇒ (P,S) ∈ R ∀ P, Q, S ∈ A

Let (P,Q) ∈ R and (Q,S) ∈ R ∀ P, Q, S ∈ A

⇒ OP = OQ and OQ = OS

⇒ OP = OS

⇒ (P,S) ∈ R

Thus, R is transitive.

Since R is reflexive, symmetric and transitive it is an equivalence relation on A.