Let O be the origin.

Question:

Let O be the origin. We define a relation between two points P and Q in a$\Rightarrow O P=O Q$ and $O Q=O R$

$\Rightarrow O P=O Q=O R$

$\Rightarrow O P=O R$

$\Rightarrow(P, R) \in R$ plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Solution:

Let A be the set of all points in a plane such that

$A=\{P: P$ is a point in the plane $\}$

Let $R$ be the relation such that $R=\{(P, Q): P, Q \in A$ and $O P=O Q$, where $O$ is the origin $\}$

We observe the following properties of R.

Reflexivity: Let P be an arbitrary element of R.

The distance of a point P will remain the same from the origin.
So, OP = OP

$\Rightarrow(P, P) \in R$

So, $R$ is reflexive on $A$.

Symmetry : Let $(P, Q) \in R$

$\Rightarrow O P=O Q$

$\Rightarrow O Q=O P$

$\Rightarrow(Q, P) \in R$

So, $R$ is symmetric on $A$.

Transitivity : Let $(P, Q),(Q, R) \in R$

$\Rightarrow O P=O Q$ and $O Q=O R$

$\Rightarrow O P=O Q=O R$

$\Rightarrow O P=O R$

$\Rightarrow(P, R) \in R$

So, $R$ is transitive on $A$.

Hence, R is an equivalence relation on A.