# Symmetric difference

• January 25th 2011, 08:07 PM
steph3824
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!
• January 25th 2011, 11:34 PM
FernandoRevilla
A possible way:

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

then

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

Prove that:

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

or

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

analyzing the cases

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

Fernando Revilla