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Homework Help: Sets: A\B can represent A union B?

  1. Jun 10, 2010 #1
    As a event A\B stands for "A occurs but B does not." Show that the operations of union, intersection and complement can all be expressed using only this operation.

    [tex]A \backslash B = A \cap \bar{B}[/tex]

    So far I have resorted to making a truth table with a bunch of A\B combinations that look at A\B, (A\B)\B, ((A\B)\B)\B), and so on. I don't see anything very interesting with this approach. What is a more logical way to look at this problem? If I could find a "nand" or "nor" combination, then I could make any operator. Do I just have to stumble on to it?

    Last edited: Jun 10, 2010
  2. jcsd
  3. Jun 10, 2010 #2
    The order I would suggest you do this in is complement, union, intersection. A logical way to organize your work is to use complement in doing union and using union to do intersection. It is important, by the way, to remember that set difference is not associative. That is, (A\B)\C is not in general equal to A\(B\C). However, A\B = (A\B)\B = ((A\B)\B)\B = ..., which you can see from thinking about the elements in those sets.

    EDIT: Also, remember that complement is really an operation involving two sets, A and the universe of discourse X.
    Last edited: Jun 10, 2010
  4. Jun 10, 2010 #3
    Thank you Tedjn. The order you suggested and the tip at the very end helped me solve this problem quickly.
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