Show that the relation R in the set A of points in a plane given by `R = {(P , Q) :`distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set o

Show that the relation R in the set A of points in a plane given by `R

1.	Show that the relation R in the set A of points in a plane given by `R = {(P , Q) :`distance of the point P from the origin is same as the distance of the point Q from the origin}, is an equivalence relation. Further, show that the set o
Answer» `R {(P,Q):` distance of point P from the origin is the same as the distance of point Q from the origin} Clearly, `(P,P) in R` since the distance of point P from the origin is always the same as the distance of the same point P from the origin. Therefore, R is reflexive. Now, let `(P,Q) in R`. ` implies` The distance of point P from the origin is the same as the distance of point Q from the origin. `implies` The distance of point Q from the origin is the same as the distance of point P from the origin. `implies (Q,P) in R` Therefore , R is symmetric. Now, let `(P,Q),(Q,S) in R.` `implies ` The distance of point P and Q from the origin is the same and also, the distance of point Q and S from the origin is the same. `rArr` the distance of points P and S from the origin is the same. ` implies (P,S) in R` Therefore, R is transitive. Therefore, R is an equivalence relation.