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 points related to a point P ≠ (0, 0) is the circle passing through P with origin as centre.

solution:

R = {(P, Q): distance of the point P from the beginning is equivalent to the distance of the point Q from the origin}

Say “O” is beginning Point.

Since the distance of the point P from the beginning is consistently equivalent to the distance of a similar point P from the beginning.

Operation = OP

So (P, P) R. R is reflexive.

Distance of the point P from the beginning is equivalent to the distance of the point Q from the beginning

Operation = OQ then OQ = OP

R is symmetric.

Likewise OP = OQ and OQ = OR then, at that point OP = OR. R is transitive. Subsequently, R is a comparable connection.