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

Explanation

Let us denote this relation `b y R={(P, Q): O P=O Q}` for `O` being the origin. Now `(P, P) in R` since `O P=O Q` for any point `P`. So the relation is reflexive. Again this relation is symmeric as if `(P, Q) in R Rightarrow(Q, P) in R` since `O P=O P Rightarrow` `O Q=O P` for all `P, Q`.


Also this relation is transitive as if `(P, Q) in R,(Q, S) in R Rightarrow(P, S) in R` since `{OP}={OQ}, {OQ}={OS} Rightarrow {OP}={OS}` for all `{P}, {Q}, {S}`.


Hence the relation is reflexive, symmetric and transitive so it is an equivalance relation.