question_answer 11)

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 of all point related to a point  Is the circle passing through P with origin as centre.  

Answer:

Let A be the set of points in a plane.       R = {(P, Q) : distance of the point P from the origin is same as the distance of the point Q from the origin}        R = {(P, Q) : |OP| = |OQ|, where Q is the origin}       Reflexive : As |OP| = |OP|              is reflexive.       Symmetric : Let (P, Q)              R is symmetric.       Transitive : Let (P, Q) |OQ|              R is symmetric.       Transitive : Let (P, Q) and (Q, S)                    R is transitive.       Hence R is an equivalence relation.       Let B be the set of points in a plane related to                    lies on a circle passing through P with centre O}.