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Math Help - how would prove this using for set operations is..... idempotent laws associa........

  1. #1
    Newbie
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    Unhappy how would prove this using for set operations is..... idempotent laws associa........

    how would i prove this using Set operations
    i.e.
    the operations i've been given are ........
    idempotent laws
    associative laws
    commutative laws
    distributive laws
    identity laws
    Involution laws
    involution laws
    complement laws
    de morgans laws
    <br /> <br />
\emptyset\;=\;((X \cup Y) \cap (X \cup Y')) \cap ((X' \cup Y) \cap (X' \cup Y')) <br />
    using set operations

    if not then atleast provide a similar example...............OR
    refer me to a website for learning this......
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  2. #2
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    Hello, npm1!


    Prove using Set operations:

    . . \bigg[(X \cup Y) \cap (X \cup Y')\bigg] \cap \bigg[(X' \cup Y) \cap (X' \cup Y')\bigg] \;=\;\emptyset

    . . \begin{array}{ccc}\bigg[(X \cup Y) \cap (X \cup Y')\bigg] \cap \bigg[(X' \cup Y) \cap (X' \cup Y')\bigg]  & \text{Given} \\ \\<br />
\bigg[X \cup (Y \cap Y')\bigg] \cap \bigg[X' \cup (Y \cap Y')\bigg] & \text{Distr.} \\ \\<br /> <br />
\bigg[X \cup\: \emptyset\bigg] \cap \bigg[X' \cup \:\emptyset\bigg] & A \cap A' \:=\:\emptyset \\ \\<br /> <br />
X \cap X' & A \cup \emptyset \:=\:A \end{array}

    . . . . . . . . . . . . . . . . \begin{array}{ccccccc}\emptyset & \qquad\qquad & \qquad\qquad\;\; &  A \cap A' \:=\:\emptyset\end{array}

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