That is, prove that A delta (B delta C) = (A delta B) delta C.
I managed a lengthy proof but only with the use of De Morgan's Laws - just checking if this is acceptable?
Here is one way to prove it, it may be similar to your way, but here goes:
By definition, we know that $\displaystyle A \Delta B=(A \backslash B) \cup (B \backslash A)$. Therefore,
$\displaystyle (A \Delta B) \Delta C = $
$\displaystyle =((A \Delta B)\backslash C) \cup (C \backslash (A \Delta B))=$
$\displaystyle =((A \Delta B)\cap (C)^c) \cup (C \cap (A \Delta B)^c)=$
$\displaystyle =[((A \backslash B) \cup (B \backslash A))\cap (C)^c]\cup[C \cap ((A \backslash B)\cup (B \backslash A))^c]=(*)$
Now we make a little digression to simplify $\displaystyle ((A \backslash B)\cup (B \backslash A))^c$ from the above expression:
$\displaystyle ((A \backslash B)\cup (B \backslash A))^c=$
$\displaystyle =((A \cap B^c) \cup (B \cap A^c))^c=$
$\displaystyle =(A \cap B^c)^c \cap (B \cap A^c)^c=$
$\displaystyle =(A^c \cup (B^c)^c) \cap (B^c \cup (A^c)^c)=$
$\displaystyle =(A^c \cup B) \cap (B^c \cup A)=$
$\displaystyle =(A^c\cap B^c) \cup (A^c\cap A) \cup (B \cap B^c) \cup (B \cap A)=$
$\displaystyle =(A^c\cap B^c) \cup (A \cap B)$, and then plug this into the equation above;
$\displaystyle (*)=[((A \cap B^c) \cup (A^c \cap B))\cap C^c]\cup[C \cap ((A^c\cap B^c) \cup (A \cap B))]=$
$\displaystyle =[(A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c)]\cup[(C \cap A^c \cap B^c) \cup (C \cap A \cap B))]=$
$\displaystyle =(A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c)\cup (A^c \cap B^c \cap C) \cup (A \cap B \cap C)$
Next, if you exchange sets in the original expression ($\displaystyle (A \Delta B) \Delta C$) in the following way: $\displaystyle A \leftrightarrow B$;$\displaystyle B \leftrightarrow C$;$\displaystyle C \leftrightarrow A$, you'll get that
$\displaystyle (B \Delta C) \Delta A=...=(A \cap B^c \cap C^c) \cup (A^c \cap B \cap C^c)\cup (A^c \cap B^c \cap C) \cup (A \cap B \cap C)$, and from this, it follows that
$\displaystyle (A \Delta B) \Delta C =(B \Delta C) \Delta A=$(and, because the symmetric difference is a symmetric operation, i.e. $\displaystyle X \Delta Y= Y \Delta X$, it follows that)=$\displaystyle A \Delta (B \Delta C)$.
Therefore, $\displaystyle (A \Delta B) \Delta C =A \Delta (B \Delta C)$
Here is a different way to prove this, or rather to reduce it to associativity of addition $\displaystyle \oplus$ modulo 2. To remind, $\displaystyle \oplus:\{0,1\}\times\{0,1\}\to\{0,1\}$; $\displaystyle x\oplus y=x+y$ if at least one of $\displaystyle x,y$ is $\displaystyle 0$, and $\displaystyle 1\oplus 1=0$.
Definition. For a set $\displaystyle A$ and an object $\displaystyle x$, let $\displaystyle in(x,A)=1$ if $\displaystyle x\in A$ and $\displaystyle in(x,A)=0$ otherwise.
Lemma. For all sets $\displaystyle A, B$ and any object $\displaystyle x$, $\displaystyle in(x,A{\scriptstyle\triangle} B)=in(x,A)\oplus in(x,B)$.
Theorem. For all sets $\displaystyle A,B,C$, $\displaystyle A{\scriptstyle\triangle}(B{\scriptstyle\triangle}C )=(A{\scriptstyle\triangle}B){\scriptstyle\triangl e}C$.
Proof. For any $\displaystyle x$, $\displaystyle in(x,A{\scriptstyle\triangle}(B{\scriptstyle\trian gle}C))=in(x,A)\oplus(in(x,B)\oplus in(x,C))$ = $\displaystyle (in(x,A)\oplus in(x,B))\oplus in(x,C)=in(x,(A{\scriptstyle\triangle}B){\scriptst yle\triangle}C)$.
hi math members my name is Paul Otuoma from Nairobi,Kenya. Am a student at the University of Nairobi.Am persuing Bachelor of Science, where am taking Double maths and Physics..Am glad to join this group its superb..Let me ask how comes (A delta B) is (A\B) U (B\A)?? our teacher taught us that, (A delta B)= ( A-B) U (B-A) ??? Please help..
$\displaystyle A-B$ and $\displaystyle A\setminus B$ are two different notations for the same thing: set difference. In fact, it is a good idea for people to define less-standard concepts and notations when they ask questions because there are so many variations of the same concepts around the world. I am talking in general, not so much about this particular thread because both $\displaystyle A-B$ and $\displaystyle A\setminus B$ are pretty standard.