So I have this question on relations, that I really cant understand. I mean, I cant understand the question to be honest.
Suppose a set $X$ of points on the plane and we "stabilize" a point $O ∈ X$. We define a relation $R$ on $X$ as follows : If $P,Q ∈ X$, then $P R Q \iff P,Q$ are equidistant from $O$.
I need to find if it is symmetric, reflective and transitive. And I have to describe the equivalence classes.
I know the definitions for symmetric, reflective and transitive. But my main problem is that I cant really understand the question with all these points! I have solved other more straightforward questions like this.
Perhaps it will help to "pin down" and make more explicit what is meant by "$PRQ \iff P$ and $Q$ are equidistant from $O$".
Let $r_p$ denote the distance between $P$ and $O$. So $r_q$ denotes the distance between $Q$ and $O$. Then we can say that $$P R Q \iff r_p = r_q$$
Reflexivity: Now, clearly, for all points $P \in X$, $r_p = r_p$. Hence we have reflexivity.
Symmetry: For all points $P, Q \in X$, suppose $PRQ$. Then $r_P = r_Q$. Trivially, that means $r_Q = r_P$. Hence $QRP$. So the relation is symmetric.
Transitivity: We need to show that for all points $P, Q, S$, $\;(PRQ \land QRS) \rightarrow PRS$. Suppose $PRQ$ and $QRS$. Then $r_p = r_q$, and $r_q = r_s$. Since equality is transitive, we conclude that it must follow that $r_p = r_s$. And hence, we indeed have that $PRS$: $R$ is indeed transitive.
As for the equivalence classes, try thinking of $r_p$ (the distance between a point P and point $O$) as the radius of a circle centered at $O$ and intersecting the point $P$. So the equivalence class $[P]$ is the set of all points whose distance from $O$ is equal to $r_p$: i.e., the circle centered at $O$ of radius $r_p$ which intersects $P$ and all other points whose distance from $O$ is equal to $r_p$.
So the equivalence classes under $R$ consists of concentric circles of radius $r \in \mathbb R$, all centered at $O$. To every point $x\in X$ we associate a radius $r_x$ which measures the distance between $x$ and $O$. All those points of distance $r_x$ from the origin $O$ belong to the equivalence class of $x$.