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Math Help - Question on set operations

  1. #1
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    Question on set operations

    is this statement true?

    (A - B) - C = (A - C) - (B - C)
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  2. #2
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    Hello, brudman!

    Is this statement true?

    . . (A - B) - C  \:=\:  (A - C) - (B - C) . . . . . yes
    Definition: .  P - Q \:=\:P \cap Q'


    The left side is:

    . . \begin{array}{cc}<br />
(A - B) - C & \text{Given} \\<br />
(A \cap B') - C & \text{d{e}f.Subtr'n}\\<br />
(A \cap B') \cap C' & \text{d{e}f.Subtr'n} \\<br />
A \cap B' \cap C' & \text{Associative} \end{array}


    The right side is:

    . . \begin{array}{cc}<br />
(A-C) - (B-C) & \text{Given} \\<br /> <br />
(A \cap C') - (B \cap C') & \text{d{e}f.Subtr'n} \\<br /> <br />
(A \cap C') \cap (B \cap C')' & \text{d{e}f.Subtr'n} \\<br /> <br />
(A \cap C') \cap (B' \cup C) & \text{DeMorgan} \\<br /> <br />
A \cap C' \cap (B' \cup C) & \text{Associative} \\<br /> <br />
A \cap \bigg[(C' \cap B') \cup (C' \cap C)\bigg] & \text{Distributive} \\<br /> <br />
A \cap \bigg[(C' \cap B') \cup \emptyset\bigg] & P \cap P \,=\,\emptyset <br />
\end{array}
    . . . . . . \begin{array}{ccccc}<br />
A \cap (C' \cap B') &&&& P \cup \emptyset \,=\,P \\<br /> <br />
A \cap (B' \cap C') &&&& \text{Commutative} \\<br /> <br />
A \cap B' \cap C' &&&& \text{Associative}<br /> <br />
\end{array}

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