Let $S = [10]$ (the set containing the integers 1,...,10). Define an equivalence relation R on S by $xRy$ $iff$ $x^2 \equiv y^2$ (mod 5) is an equivalence relation and determine the equivalence classes.
What I have
I'll define R by $$R=\{(3,2), (4,1), (6,1), (6,4), (7,2), (7,3), (8,2), (8,3), (8,7), (9,1), (9,4), (9,6), (10,5)\}$$
Definition of equivalence classes: Given an equivalence relation of S, the set of elements equivalent to $x \in S$ is the equivalence class containing x.
I interpret this as meaning that to be in the same class that one of the coordinates must be the same as another one of the coordinates in a different ordered pair?
If so then the equivalence classes are:
$EC_1=\{(3,2), (7,2), (7,3), (8,2), (8,3), (8,7)\}$
$EC_2=\{(4,1), (6,1), (6,4), (9,1), (9,4), (9,6)\}$
$EC_3=\{(10,5)\}$
I cannot really tell if I've applied all the definitions and concepts correctly, and would appreciate any input, but this is as far as I could get from reading the chapter, in my textbook. Thank you in advanced.