 Hello and welcome to the session. Let us discuss the following problem today. Show that the relation r in the set A of points in a plane given by r is equal to p,q so that distance of the point p from the origin is same as the distance from the point q from the origin is an equivalence relation. Further, show that the set of all points related to the point p is not equal to 0,0 in the circle passing through p with origin as center. Now let us discuss the solution. Let o denote the origin of the given plane. Then r is equal to p,q so that o p is equal to o q. Now let us check for reflexivity. For any point p in the set A, p is equal to o p which implies p,p belongs to r for all p belongs to a. Therefore, r is reflexive. Now let us check for symmetry. Let p and q be two points in the set A because A is equal to points in a plane. p,q belongs to r by definition of r which implies o p is equal to o q, which implies o q is equal to o p which implies q p belongs to r. Thus, p,q belongs to r which implies q p belongs to r for all p,q belongs to a. Therefore, r is symmetric. Now let us check for transitivity. Let p,q and s be three points in the set A. Then p,q belongs to r and q,s belongs to r which implies o p is equal to o q and o q is equal to o s which implies o p is equal to o s which implies p,s belongs to r. Therefore, r is transitive. r is reflexive, r is symmetric and r is transitive. Hence, r is an equivalence relation. Now let us discuss the second part of our problem. Let p be a fixed point in set A and q be another fixed point such that p,q belongs to r which implies o p is equal to o q which implies q moves in the plane in such a way that its distance from the origin that is 0,0 is always the same and is equal to o p which implies locus of q is a circle whose center is the origin and radius is o p. Thus, set of all points related to p is the circle passing through p and its center is the origin. I hope you understood this problem. Bye and have a nice day.