Let a be the set of all points in a plane and o be the origin show that the relation to={ p, q }:p Q A and op = OQ } is an equivalence relation

Let us denote this relation by R={(P, Q): OP=OQ} for O is the origin. Now (P, P)∈R since OP=OQ for any point P. So the relation is reflexive. Again this relation is symmetric as if (P, Q)∈R⇒(Q, P)∈R since OP=OP⇒OQ=OP for all P, Q. Also, this relation is transitive as if (P, Q)∈R,(Q, S)∈R⇒(P, S)∈R since OP=OQ, OQ=OS⇒OP=OS for all P, Q, S.

Hence the relation is an equivalence relation.