<s>[INST] Discrete math $A\,\triangle\, B = C$ implies that $A\,\triangle\, C = B$ $A\,\triangle\, B = C$ implies that $A\,\triangle\, C = B$
I understand that the delta is the symmetric difference and that the symmetric difference of $A$ and $B$ is the set of elements that belong to exactly one of $A$ and $B$. How do I prove the above statement?
 [/INST] We need to prove that given $C = A \ \Delta \ B$, then $B = A \ \Delta \ C$

A simple and elementary proof is to just use the definition of the symmetric difference, or $XOR$ as I like to call it, and setup some truth tables:
$\begin{array}{|c|c|c|} \hline
    A & B & \mathbf{C} \\ \hline
    0 & 0 & \mathbf{0} \\ \hline
    0 & 1 & \mathbf{1} \\ \hline
    1 & 0 & \mathbf{1} \\ \hline
    1 & 1 & \mathbf{0} \\ \hline
\end{array}$ 
$\begin{array}{|c|c|c|c|} \hline
    A & B & \mathbf{C} & \mathbf{A \ \Delta \ C} \\ \hline
    0 & 0 & \mathbf{0} & \mathbf{0}\\ \hline
    0 & 1 & \mathbf{1} & \mathbf{1}\\ \hline
    1 & 0 & \mathbf{1} & \mathbf{0}\\ \hline
    1 & 1 & \mathbf{0} & \mathbf{1}\\ \hline
\end{array}$ 
In the last truth table we can clearly see that $B = A \ \Delta \ C$, completing our proof.
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