Etydhv

Let $A, B$ subsets of $X$ and $mathbb P(X)$ the power set,
we define the following equivalence relation on $mathbb P(X)$:

Let $ Ssubseteq 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!

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.

We need to show theat $Asim B$ 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.

If $A Delta B subseteq S$ and $B Delta C subseteq S$, then

$A Delta C= (A Delta B) Delta (B Delta C) subseteq C$ as well.

Hint:

By definition, $Asim B$ means $A-B$ and $B-Asubset S$. So you have to show that, if $A-B, B-A, B-C, C-Bsubset S$, then both $A-C$ and $C-A$ are subsets of $S$.

Consider first an element $xin A-C$. Either it is in $B$, or it is not in $B$. What can you deduce from the hypotheses in each case?