On Mon, 6 Jun 2011 21:17:51 -0700 (PDT), "email@example.com" <firstname.lastname@example.org> wrote in <news:email@example.com> in alt.math.undergrad:
> I need help with the following (not a hw problem, just > reading a book on prob theory):
> Let * be the symmetric difference operation: A * B = > (A-B) U (B-A)
> solve for X: A * X = B
> I can solve it using a venn diagram: X = A * B
> But how do I get the solution using set algegra?
The most elegant approach is to prove that * is associative and that S * S = Ø and Ø * S = S for any set S: then you have X = Ø * X = (A * A) * X = A * (A * X) = A * B.
The only hard part of that is the associativity of *, which is probably most easily proved by element chasing (i.e., by showing that x is a member of A * (B * C) if and only if x is a member of (A * B) * C.