Let $A, B$ subsets of $X$ and $\mathbb P(X)$ the power set, we define the following equivalence relation on $\mathbb P(X)$:
Let $ S\subseteq X$ a fixed subset of $X$ and $A$~$B$ $\iff A△B \subseteq S$
Prove that is is an equivalence relation and find the class of $X$ and $S$
I have already shown that the relationship satisfies reflexivity and symmetry, all this is justified respectively by the fact that the empty set is a subset of any set and the symmetric difference is commutative.
My problem is with transitivity, I do not know how to do it, that is when I try to use it for the definition of symmetric difference I fall in many cases. There is some way to test transitivity using only operations between sets. And with respect to the equivalence class of $S$, I showed that they are all subsets of $X$ contained in $S$. But with respect to the equivalence class of $X4 I do not see what it is.
Any help would be useful. Thank you!