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 ne (0,0)` is the circle passing through P with origin as centre.

Explanation

`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.