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Re: symmetric difference calculation
Posted:
Jun 7, 2011 8:43 PM


In article <d3f809e5f9b3477ab82edd75689cc017@f31g2000pri.googlegroups.com>, <"g_chime@yahoo.com"> wrote:
> 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 = (AB) U (BA) > > 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? > > Thanks in advance.
There is also a very inelegant pointwise argument  it helps to know the answer ahead of time.
Let x be in X. If x is not in A then x is in X  A thus x is in B so x is in B  A. If x is in A then x is not in (A  X) \/ (X  A) = B so x is in A  B. In either case x is in (A  B) \/ (B  A).
On the other hand, let x be in (A  B) \/ (B  A) and note that x is in A \/ X. Suppose x is in A. Then x is not in B so x is in X or else x is in A  X which is a subset of B. If x is not in A then x is in B  A. Since x is in B and x is not in A  X it follows that x is in X  A and consequently in X.
 Paul Sperry Columbia, SC (USA)



