Equivalence relation by the symmetric difference of sets 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 $S$
My work:
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!
 A: If $A \Delta B \subseteq S$ and $B \Delta C \subseteq S$, then
$A \Delta C= (A \Delta B) \Delta (B \Delta C) \subseteq S$ as well.
A: We need to show theat $A\sim B$ and $B \sim C$ give $A \sim C$. This can easily be seen by visualising $A, B, C$ in a Venn diagram (try it yourself!)
To put the Venn diagram proof formally, consider any element $x$ in $A$ but not in $C$. If $x$ is in $B$, it lies in the symmetric difference of $B$ and $C$ and so is in $S$. If $x$ is not in $B$, it lies in the symmetric difference of $A$ and $B$ and so is in $S$. By symmetry any element in $C$ but not $A$ is in $S$, completing the proof.
A: Hint:
By definition, $A\sim B$ means $A-B$ and $B-A\subset S$. So you have to show that, if $A-B, B-A, B-C, C-B\subset S$, then both $A-C$ and $C-A$ are subsets of $S$.
Consider first an element $x\in A-C$. Either it is in $B$, or it is not in $B$. What can you deduce from the hypotheses in each case?
A: Two hints about ways to go.


*

*Draw a Venn diagram for $A, B, C, X$ showing $A \Delta B$ and so on.

*Know or show that $\Delta$ is associative. That and the fact that $B
   \Delta B$ is empty leads to an algebraic proof.

A: Below, a proof using the fact that (1) symmetrixc difference can be defined using exclusive OR ( w) (2) that exclusive OR is the negation of the biconditional operator, and (3) that the biconditional operator is transitive
Below, please read " we'll " instead of " will". 

