# Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation. 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

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