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Math Help - Complete sets demonstrations

  1. #1
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    Complete sets demonstrations

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

    Demonstrate that A_1, A_2 are complete sets too. And if A is a complete set then 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:
    A \cap \bigcap X \neq \emptyset \rightarrow A^c \cap (\bigcap X)^c \neq \emptyset with \bigcap X = A_1^{i1} \cap A_2^{i2} then 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 A\subseteq \mathcal{U}. For i_1, i_2 \in \{0,1\}and A^0 := A^c, A^1 := A.
    A is a complete set if A\cap A_1 ^{i1} \cap A_2^{i2} \neq \emptyset then A_1 ^{i1} \cap A_2^{i2} \subseteq A

    Demonstrate that A_1, A_2 are complete sets too. And if A is a complete set then 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:
    A \cap \bigcap X \neq \emptyset \rightarrow A^c \cap (\bigcap X)^c \neq \emptyset with \bigcap X = A_1^{i1} \cap A_2^{i2} then 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 A_1,A_2?
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