Question

Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.

Answer 1

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

$$\begin{gathered} A=\{P:P\mathrm{is}\mathrm{a}\mathrm{point}\mathrm{in}\mathrm{the}\mathrm{plane}\} \\ \mathrm{Let}R\mathrm{be}\mathrm{the}\mathrm{relation}\mathrm{such}\mathrm{that}R=\left\{\left(P,Q\right):P,Q\in A\mathrm{and}OP=OQ,\mathrm{where}O\mathrm{is}\mathrm{the}\mathrm{origin}\right\} \end{gathered}$$

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
$$\begin{gathered} \Rightarrow \left(P,P\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{reflexive}\mathrm{on}A. \\ \mathrm{Symmetry}:\mathrm{Let}\left(P,Q\right)\in R \\ \Rightarrow OP=OQ \\ \Rightarrow OQ=OP \\ \Rightarrow \left(Q,P\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{symmetric}\mathrm{on}A. \\ \mathrm{Transitivity}:\mathrm{Let}\left(P,Q\right),\left(Q,R\right)\in R \\ \Rightarrow OP=OQ\mathrm{and}OQ\mathit{=}OR \\ \Rightarrow OP=OQ=OR \\ \Rightarrow OP=OR \\ \Rightarrow \left(P,R\right)\in R \\ \mathrm{So},R\mathrm{is}\mathrm{transitive}\mathrm{on}A. \end{gathered}$$

Hence, R is an equivalence relation on A.