Let $R$ be an equivalence relation over $P(\{1,2,3\})$, defined as following:
$$\forall A,B \in P(\{1,2,3\}): \ (A,B)\in R \iff \ A \triangle B \subseteq \{1,2\}$$
as $\triangle$ represents symmetric difference [$A\triangle B = (A\cup B)\setminus(A\cap B)$].
It's no problem to understand for which $A, B \in P(\{1,2,3\}): (A,B)\in R$, and also to check for which $A, B \in P(\{1,2,3\}) (A,B)\notin R$.
My problem is to find all equivalence classes in a "mathematical" way -- meaning to understand how to find them and not by "guessing".
The relation $R$ can be described as follows: $(A,B) \in R$ iff either both $A$ and $B$ contain $3$ or neither contains $3$. Thus, for any $A$ you just have to consider two cases:
a) $3 \in A$ and
b) $3 \notin A$
In case a) the equivalence class consists of all sets that contain $3$ and in case b) the equivalence class consists of all sets that do not contain $3$.