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Math Help - Boolean algebra / venn diagrams

  1. #1
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    Boolean algebra / venn diagrams

    For any two subsets A and B of some set S,

     (A \cup B) - (A \cap B) = (A - B) \cup (B - A)

    where  X-Y = X \cap Y'

    Show that this is true by using a) Venn diagrams and b) algebraic manipulation using the properties of Boolean algebra.



    Ok so im struggling to start on this. Part a i don't know how to start really. Part b I'm tempted to write

    X - Y = (X - Y')\cup(Y' - X)

    but doubt its right :/
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  2. #2
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    Draw two circles which intersect in a Venn diagram, label one A and the other B. Figure out the corresponding regions represented by the each expression an it should become clear.

    For b) use distributivity and De Morgan's:

    (A \cup B) \cap (A \cap B)^c=(A \cup B) \cap (A^c \cup B^c) <br />
= [(A \cup B) \cap A^c] \cup [(A \cup B) \cap B^c]<br />
= (B \cap A^c) \cup (A \cap B^c)
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