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

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Brian M. Scott

Posts: 1,289
Registered: 12/6/04
Re: symmetric difference calculation
Posted: Jun 7, 2011 1:06 AM
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On Mon, 6 Jun 2011 21:17:51 -0700 (PDT), ""
<> wrote in
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.


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