1. 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

2. Re: Complete sets demonstrations

Originally Posted by nestora
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 $\displaystyle A_1,A_2$?