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Math Help - Union and Intersection proof

  1. #1
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    Union and Intersection proof

    For a set of Y real numbers define a function CY by;

    CY(x)= {1 if x ∈Y,
    0 if x is not ∈ Y.

    Let Y and Z be a set of real numbers, find expressions for CYUZ and Cynz
    YUZ={x∈R|x∈Y or x∈Z}
    ynz={x∈R|x∈Y and x∈Z}

    so i know what Cy is but I'm unsure of what Cz is and what I am suppose to do. Help would be greatly appreciated.
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: Union and Intersection proof

    Using the given definitions, you'll obtain

    (i)\;\;\;C_{Y\cap Z}=C_Y\cdot C_Z
    (ii) \;\;C_{Y\cup Z}=C_Y+C_Z-C_Y\cdot C_Z

    For example:

    x\in Y\cap Z\Rightarrow x\in Y\mbox{ and } x \in Z \Rightarrow \begin{Bmatrix} C_{Y\cap Z}(x)=1\\C_{Y}(x)=1\\C_{Z}(x)=1\end{matrix}\\ \Rightarrow C_{Y\cap Z}(x)=C_Y(x)\cdot C_Z(x)

    x\not\in Y\cap Z\Rightarrow \ldots

    (try the rest).
    Last edited by FernandoRevilla; September 30th 2012 at 10:25 PM.
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  3. #3
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    Re: Union and Intersection proof

    Quote Originally Posted by FernandoRevilla View Post
    Using the given definitions, you'll obtain

    (i)\;\;\;C_{Y\cap Z}=C_Y\cdot C_Z
    (ii) \;\;C_{Y\cup Z}=C_Y+C_Z-C_Y\cdot C_Z
    Alternatively,

    C_{Y\cap Z}(x)=\min(C_Y(x), C_Z(x))
    C_{Y\cup Z}(x)=\max(C_Y(x),C_Z(x)).
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