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 , suppose (without loss of generality) that

then



Prove that:



or



analyzing the cases

.



Fernando Revilla