Question

On the set of all points in a plane, the relation defined by the phrase 'at the same distance from the origin ' is an equivalence relation.

• A. True
• B. False

Answer 1

The correct option is A True

True.

Consider an example:

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): 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 $(P,Q)\in R$.

The distance of point P from the origin is the same as the distance of point Q from the origin.

The distance of point Q from the origin is the same as the distance of point P from the origin.

$(Q,P)\in R$

Therefore R is symmetric.

Now,

Let $(P,Q),(Q,S)\in R$.

The distance of points P and Q from the origin is the same and also, the distance of points Q and S from the origin is the same.

The distance of points P and S from the origin is the same.

$(P,S)\in R$

Therefore R is transitive.

Therefore, R is an equivalence relation.

The set of all points related to $P\ne (0,0)$ will be those points whose distance from the origin is the same as the distance of point P from the origin.

In other words, if O (0, 0) is the origin and OP = k, then the set of all points related to P is at a distance of k from the origin.