Symmetric difference is associative?
Is the symmetric difference associative? I'd like to be able to give a proof for it.
I.e. is the following true? , where A, B, and C are sets.
To me it seems this is the case, i.e. that the symmetric difference is associative.
Let \ . I.e. assume that there is an element a in that is not in
This implies that a must be in only one of A, B or C for this to be true.
But if a is in either A, B or C, but not in two or more; then by definition of the symmetric difference, it must also be in Which is a contradiction -- a cannot be in without also being in
But how can I give a proof for this?