prove that for sets $\displaystyle A,B,C,D$
$\displaystyle A \Delta B = C \Delta D$ if and only if $\displaystyle A \Delta C = B \Delta D$
There is also a different way to solve this that sometimes works very well with symmetric difference.
Define the function $\displaystyle in$ by $\displaystyle in(x,A)=1$ if $\displaystyle x\in A$ and $\displaystyle in(x,A)=0$ if $\displaystyle x\notin A$. Here $\displaystyle A$ is any set and $\displaystyle x$ is any object.
Also, define the function $\displaystyle \oplus:\{0,1\}\times\{0,1\}\to\{0,1\}$ as follows:
$\displaystyle 0\oplus0=0$
$\displaystyle 0\oplus1=1$
$\displaystyle 1\oplus0=1$
$\displaystyle 1\oplus1=0$
(If we associate 1 with True and 0 with False, then $\displaystyle x\oplus y=\neg(x\leftrightarrow y)$. Also, $\displaystyle \oplus$ is addition modulo 2.) One can check that $\displaystyle \oplus$ is commutative and associative.
Show that $\displaystyle in(x,A\triangle B)=in(x,A)\oplus in(x,B)$ for all $\displaystyle A$, $\displaystyle B$, and $\displaystyle x$.
The part above was generic; it applies to any problem with symmetric difference and has to be done only once.
Now, from this problem's assumption and the fact above we have $\displaystyle in(x,A)\oplus in(x,B)=in(x,C)\oplus in(x,D) $ for all $\displaystyle x$. We can add (using $\displaystyle \oplus$) an expression $\displaystyle in(x,E)$ to both sides where $\displaystyle E$ is $\displaystyle A$, $\displaystyle B$, $\displaystyle C$, or $\displaystyle D$. E.g., if we add $\displaystyle in(x,B)$ to both sides, we get $\displaystyle in(x,A)$=$\displaystyle in(x, C)\oplus in(x,D)\oplus in(x,B)$. From here it is easy to show the desired equation.
Hello santiagos11I think this is easiest to prove by re-writing it as the equivalent statement using logical propositions. In other words:Prove that if $\displaystyle a, b,c, d$ are logical propositions, then $\displaystyle a \oplus b \equiv c \oplus d$ if and only if $\displaystyle a \oplus c \equiv b \oplus d$, where $\displaystyle \oplus$ denotes 'exclusive or'.This is quite straightforward if you use a Truth Table for each of the compound propositions and show that each table produces an identical output.
The attached diagram shows the truth table for $\displaystyle a \oplus b \equiv c \oplus d$. It is a simple matter to show that the table for $\displaystyle a \oplus c \equiv b \oplus d$ produces the same result.
Grandad