On the Wikipedia page about Equivalence Relations, there is a simple example:
Let the set $\{a,b,c\}$ have the equivalence relation $\{(a,a),(b,b),(c,c),(b,c),(c,b)\}$. The following sets are equivalence classes of this relation:
$[a]=\{a\},~~~~~[b]=[c]=\{b,c\}$.
The set of all equivalence classes for this relation is $\{\{a\},\{b,c\}\}$.
I don't understand why $\{c,b\}$ is not an equivalence class as well as $\{b,c\}$.
Furthermore, if $[a]=\{a\}$, then shouldn't the same hold for $[b]=\{b\}$ and $[c]=\{c\}$ since $\{b,b\}$ and $\{c,c\}$ are both in the relation?
Edit
After more thought, I've realised that my confusion may be to do with not understanding the notation properly. Is the Wikipedia article stating the $b$ and $c$ are equivalent (which would mean that $\{b,c\}$ and $\{c,b\}$ are the same)?
The curly braces denote sets, so the order does not matter, ie. $\{b,c\}=\{c,b\}$. There is a difference between an equivalence relation and the equivalence classes. The relation is an ordered pair $(a,b)$, which means that $a$ and $b$ are equivalent.
The equivalence class is the set of all equivalent elements, so in your example, you have $[b]=[c]=\{b,c\}=\{c,b\}$. But it is not true that $[b]=\{b\}$, since $\{b\}\neq\{b,c\}$ (instead you could write $b\in [b]$).