Discrete Mathematics Symmetric Diffirence Proof I've been trying to find a proof for the following problem but have been unable to come up with anything myself:
Say we have A, B, C part of a universe U
show that if $$A \Delta C = B \Delta C \rightarrow A = B$$
I've been able to use set operations to change the form of the problem into a logics one:
$${x | X \in A \cup C \vee x \notin A \cap C}  $$
$${x | x \in A \vee x \in C \vee x \notin (x \in A \wedge x \in C)}$$
A second thought was using the definition of symmetric diffirence
where $$A \Delta C = (A \cup C) - (A \cap C)$$
Can anyone provide a step by step proof or point me in the right direction?
 A: I feel like this is close to a proof by contradiction but don't feel it is formal enough yet looking for help on fixing it: 


*

*symmetric diffirence is the set of elements which are in either of the sets and not in their intersection.  

*the intersection of two sets is the set that contains all elements that are present in both sets.   

*so say I have an $$ x \in ( A \Delta C ) $$ and $$x \in ( A \Delta C )$$ then A = B because there can not be two symmetric differences of C that are the same unless A = B
A: I think the nicest way is to first prove


*

*$A\triangle (B\triangle C) = (A\triangle B)\triangle C$

*$A\triangle A = \varnothing$

*$A\triangle\varnothing = A$


Then applying $\triangle C$ to both sides of your hypothesis quickly yields the conclusion.
(That is, start with
$ A\triangle C = B\triangle C
\implies (A\triangle C)\triangle C = (B\triangle C)\triangle C $.)
A: An alternative answer inspired by Steven Taschuk:
Given that:


*

*$A\Delta (B\Delta C) = (A\Delta B)\Delta C$

*$ A\Delta C = B\Delta C$  

*$ A \Delta \emptyset = A$

*$ A \Delta A = \emptyset$


The proof follows:
$$ A \Delta \emptyset = A$$
$$ \leftrightarrow$$
$$ A \Delta (C\Delta C) $$
$$ \leftrightarrow$$
$$ (A \Delta C) \Delta C $$
$$ \leftrightarrow$$
$$ (B \Delta C) \Delta C $$
$$ \leftrightarrow$$
$$ B \Delta (C\Delta C) $$
$$ \leftrightarrow$$
$$ B \Delta \emptyset $$
$$ \leftrightarrow$$
$$ B $$
Feedback on correctness is greatly appreciated.
