Basic Equivalence Class Discrete Math

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: $$[0]=[3]=\{0,3\} \\ [1]=[4]=\{1,4\} \\ [2]=[5]=\{2,5\}$$

maybe this example i found can help: If X is the set of all cars, and ~ is the equivalence relation "has the same color as", then one particular equivalence class consists of all green cars. X/~ could be naturally identified with the set of all car colors.