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Thread: 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,

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

    where $\displaystyle 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

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

    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 $\displaystyle A$ and the other $\displaystyle B$. Figure out the corresponding regions represented by the each expression an it should become clear.

    For b) use distributivity and De Morgan's:

    $\displaystyle (A \cup B) \cap (A \cap B)^c=(A \cup B) \cap (A^c \cup B^c)$$\displaystyle
    = [(A \cup B) \cap A^c] \cup [(A \cup B) \cap B^c]
    = (B \cap A^c) \cup (A \cap B^c)$
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