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Thread: Simplify using the set rules of inference

  1. #1
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    Simplify using the set rules of inference

    Simplify the following: A, B, and C are subsets of the Universe.

    $\displaystyle \overline{[((A-B)\cup(A \cap C)) \cap\overline{(A \cap B)}}$
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  2. #2
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    Hello, aaronrj!

    I think I've got it . . .


    $\displaystyle A, B,C$ are subsets of the Universe.

    Simplify: .$\displaystyle \overline{\left[\,(A-B)\cup(A \cap C)\,\right] \:\cap \:\overline{(A \cap B)}}$
    Under the long negation bar, we have:

    $\displaystyle \left[\,(A - B) \cup (A \cap C)\,\right] \:\cap \:\overline{(A \cap B)}$

    . . $\displaystyle =\; \left[\,(A \cap \overline{B}) \cup (A \cap C)\,\right] \:\cap\:(\overline{A} \cup \overline{B}) $ . . Def. of set subtraction, DeMorgan's Law

    . . $\displaystyle =\;\left[\,A \cap (\overline{B} \cup C)\,\right] \cap (\overline{A} \cup \overline{B}) $ . . . . . . Distributive Property

    . . $\displaystyle =\;A \cap \left[\,(\overline{B} \cup C) \cap (\overline{A} \cup \overline{B})\,\right]$ . . . . . . Associative Property

    . . $\displaystyle =\; A \cap \left[\,(\overline{B} \cup C) \cap (\overline{B} \cup \overline{A})\,\right]$ . . . . . . Commutative Property

    . . $\displaystyle =\;A \cap \left[\,\overline{B} \cup (C \cap \overline{A})\,\right]$ . . . . . . . . . .Distributive Property

    . . $\displaystyle =\; (A \cap \overline{B}) \,\cup \, \left[\,A \cap (C \cap \overline{A})\,\right]$ . . . . . .Distributive Property

    . . $\displaystyle =\;(A \cap \overline{B}) \,\cup\,\underbrace{\left[\,A \cap \overline{A} \cap C\,\right]}_{\text{This is false}}$ . . . . . . .Comm., Assoc. Properties

    . . $\displaystyle =\qquad A \cap \overline{B}$ . . . . . . . . . . . . . . .
    Don't know the name of this property.


    Replacing the long negation bar, we have:

    . . . . $\displaystyle \overline{(A \cap \overline{B})} \;\;=\;\;\overline{A} \cup B$ . . . . . . . . . DeMorgan's Law

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