TITLE: Not understanding the concept of equivalence class
QUESTION [7 upvotes]: Let $U$ be a set defined: $U=\{(x,y)\in \Bbb R^2\mid x^2+y^2=1; xy\neq 0\}$, and let $R$ be relation defined: $(x_1,y_1)R(x_2,y_2) \iff (x_1 \cdot x_2>0∧y_1\cdot y_2>0)$. 
I was to prove it's an equivalence relation - which I did. Then I was asked to show its equivalence classes. So, I don't really understand what it is, and I don't understand how to write it with proper notation. 
Thanks for any help! 
edit: Sorry for all the typos!

REPLY [12 votes]: ${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$

REPLY [3 votes]: The equivalence classes of $R$ are the sets $C_i$ such that $a,b\in C_i \implies aRb$. Another way of thinking about this is to take a single 'representative' member $c_i$ of each class, and then $C_i \equiv [c_i] = \{a: c_iRa\}$, the set of all things related to $c_i$.
Let's start by picking $c_1=(1/\sqrt 2,1/\sqrt 2)$ to the top right of the circle. Assuming the condition is the product of the $x$ coordinates with each other is positive and similarly the $y$ coordinates: $c_1R(x,y) \iff x>0 \text{ and } y>0$. Therefore $C_1 = \text{part of circle in top right quadrant} $. Can you find the other classes? Just pick points you haven't yet put into a class.