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Thread: Complete sets demonstrations

  1. #1
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    Complete sets demonstrations

    1. We have $\displaystyle A\subseteq \mathcal{U}$. For $\displaystyle i_1, i_2 \in \{0,1\}$and $\displaystyle A^0 := A^c, A^1 := A$.
    A is a complete set if $\displaystyle A\cap A_1 ^{i1} \cap A_2^{i2} \neq \emptyset$ then $\displaystyle A_1 ^{i1} \cap A_2^{i2} \subseteq A$

    Demonstrate that $\displaystyle A_1, A_2$ are complete sets too. And if A is a complete set then $\displaystyle A^c$ is a complete set too.



    2. The first part I can't get it and don't know where to begin. The second part I tried to do:
    $\displaystyle A \cap \bigcap X \neq \emptyset \rightarrow A^c \cap (\bigcap X)^c \neq \emptyset$ with $\displaystyle \bigcap X = A_1^{i1} \cap A_2^{i2}$ then $\displaystyle A^c \subseteq (\bigcap X)^c \rightarrow A^c \subseteq (A_1^{i1})^c \cup (A_2^{i2})^c$. But when I get there I'm not sure where to go next.

    Any help would be very apreciated! Thanks
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Complete sets demonstrations

    Quote Originally Posted by nestora View Post
    1. We have $\displaystyle A\subseteq \mathcal{U}$. For $\displaystyle i_1, i_2 \in \{0,1\}$and $\displaystyle A^0 := A^c, A^1 := A$.
    A is a complete set if $\displaystyle A\cap A_1 ^{i1} \cap A_2^{i2} \neq \emptyset$ then $\displaystyle A_1 ^{i1} \cap A_2^{i2} \subseteq A$

    Demonstrate that $\displaystyle A_1, A_2$ are complete sets too. And if A is a complete set then $\displaystyle A^c$ is a complete set too.



    2. The first part I can't get it and don't know where to begin. The second part I tried to do:
    $\displaystyle A \cap \bigcap X \neq \emptyset \rightarrow A^c \cap (\bigcap X)^c \neq \emptyset$ with $\displaystyle \bigcap X = A_1^{i1} \cap A_2^{i2}$ then $\displaystyle A^c \subseteq (\bigcap X)^c \rightarrow A^c \subseteq (A_1^{i1})^c \cup (A_2^{i2})^c$. But when I get there I'm not sure where to go next.

    Any help would be very apreciated! Thanks
    This is very confusing. What are $\displaystyle A_1,A_2$?
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