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Topic: symmetric difference calculation
Replies: 3   Last Post: Jun 14, 2011 12:34 PM

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Paul Sperry

Posts: 1,392
Registered: 12/6/04
Re: symmetric difference calculation
Posted: Jun 7, 2011 8:43 PM
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In article
<""> 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 = (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?
> Thanks in advance.

There is also a very inelegant point-wise 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)

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