TITLE: Basic Equivalence Class Discrete Math
QUESTION [9 upvotes]: I read through the textbook definition of the equivalence class, but still cannot clearly understand what an equivalence class is. 
Does anyone have a good example with a definition that can hit me home?

REPLY [18 votes]: An equivalence class is just a set of things that are all "equal" to each other. Consider the set
$$S=\{0,1,2,3,4,5\}.$$
There are many equivalence relations we could define on this set. One would be $xRy \Leftrightarrow x=y$, in which case the equivalence classes are:
$$[0]=\{0\} \\ [1]=\{1\} \\ \vdots \\ [5]=\{5\}$$
We could also define $xRy$ if and only if $x \equiv y \pmod{3}$, in which case our equivalence classes are:
$$[0]=[3]=\{0,3\} \\ [1]=[4]=\{1,4\} \\ [2]=[5]=\{2,5\}$$