0

Consider the following relation $\mathfrak{R}$: in a plane $\pi$, fixed a point $O$, for every pair of points $A$ and $B$, we say that $A \mathfrak{R} B$ if and only if $A,B,O$ are collinear. Is $\mathfrak{R}$ an equivalence relation? If yes, find the equivalence classes.

I have estabilished that $\mathfrak{R}$ is an equivalence relation and I have found the equivalence classes: they are the lines passing through the point $O$.

But now I have a problem: the point $O$ belongs to every equivalence class, because it belongs to every line passing through the point $O$. This is a problem because the equivalence classes have to be disjoint.

What am I doing wrong? How is it possible?

user1988
  • 47
  • Are you sure, this relation is transitive ? – hamam_Abdallah Aug 20 '21 at 21:08
  • I would say yes. If $A,O,B$ are collinear, than $B$ belongs to the line $AO$ and the line $BO$ is the same as $AO$. If $B,O,C$ are collinear, then $C$ belongs to the line $BO$, but $BO$ is the same as $AO$, so $C$ belongs to the line $AO$ and $A,O,C$ are collinear. – user1988 Aug 20 '21 at 21:17
  • 1
    You can find A and B such that $ A R O $ and $ O R B $ but $ O,A,B $ not colinear. – hamam_Abdallah Aug 20 '21 at 21:20
  • How? If $A,R,O$ are collinear, then $A$ belongs to the line $RO$. If $O,R,B$ are collinear, then $B$ belongs to the line $RO$. So $A$ and $B$ belong to the $RO$ and $A,O,B$ are collinear! – user1988 Aug 20 '21 at 21:29
  • By $ A R O$ i mean $A$ is in relation with $ O $ – hamam_Abdallah Aug 20 '21 at 21:34
  • Oh my G*d! You are right! I'm an idiot! Thanks. – user1988 Aug 20 '21 at 21:39

0 Answers0