Symmetric difference

Please forgive me in advance if this post is not in the correct section!! Anyways, one of my homework problems is to show that the symmetric difference Δ satisfies the cancellation law, namely, if AΔB=AΔC then B=C. It seems like this is fairly simple however I just can't seem to figure out what to do with this. Thank you for your help!

A possible way:

If $\displaystyle B\neq C$ , suppose (without loss of generality) that $\displaystyle B\not\subset C$

then

$\displaystyle \exists x: (x\in B\;\wedge\;x\notin C)$

Prove that:

$\displaystyle A \Delta C\not\subset A\Delta B$

or

$\displaystyle A \Delta B\not\subset A\Delta C$

analyzing the cases

$\displaystyle x\in A\;\textrm{or}\;x\notin A$.



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