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Section 3.3 Families of Sets
Definition 3.30 .
Let \(A\) be a set. The set \(A\) is called a family of sets if all the elements of A are sets. The family of sets \(A\) is indexed by \(I\text{,}\) denoted \(A = \{A_i\}_{i\in I}\text{,}\) if there is a non-empty set \(I\) such that there is an element \(A_i\in A\) for each \(i\in I\text{,}\) and that every element of \(A\) equals \(A_i\) for exactly one \(i\in I\text{.}\)
Definition 3.32 .
Let \(A\) be a family of sets. The union of the sets in \(A\text{,}\) denoted \(\bigcup_{X\in A} X\text{,}\) is defined as follows. If \(A \neq \emptyset\text{,}\) then
\begin{equation*}
\bigcup {X\in A} X = \{x | x\in A \text{ for some }A\in A\};
\end{equation*}
if \(A = \emptyset\text{,}\) then \(\bigcup{X\in A} X = \emptyset\text{.}\) The intersection of the sets in \(A\text{,}\) denoted \(\bigcap{X\in A} X\text{,}\) is defined as follows. If \(A \neq\emptyset\text{,}\) then
\begin{equation*}
\bigcap {X\in A} X = \{x | x\in A \text{ for all } A\in A\};
\end{equation*}
if \(A = \emptyset\text{,}\) then \(\bigcap{X\in A} X\) is not defined. If \(A = \{A_i\}i\in I\) is indexed by a set \(I\text{,}\) then we write
\begin{equation*}
\bigcup {i\in I}A_i = \{x | x\in A_i \text{ for some }i\in I\} \text{ and } \bigcap{i\in I}A_i = \{x | x\in A_i \text{ for all } i\in I\}
\end{equation*}
to denote the union and intersection of the sets in \(A\text{,}\) respectively.
Intuitively, the set \(\bigcup i\in I A_i\) is the set that contains everything that is in at least one of the sets \(A_i\text{;}\) the set \(\bigcap i\in I A_i\) is the set containing everything that is in all of the sets \(A_i\text{.}\) The same holds for the non-indexed notation.
Theorem 3.33 .
Let \(I\) be a non-empty set, let \(\{A_i\}_{i\in I}\) be a family of sets indexed by \(I\) and let \(B\) be a set.
\(\bigcap_{i\in I} A_i \subseteq A_k\) for all \(k\in I\text{.}\) If \(B \subseteq A_k\) for all \(k\in I\text{,}\) then \(B \subseteq\bigcap_{i\in I} A_i\) .
\(A_k \subseteq\bigcup_{i\in I} A_i\) for all \(k\in I\text{.}\) If \(A_k \subseteq B\) for all \(k\in I\text{,}\) then \(\bigcup_{i\in I} A_i \subseteq B\text{.}\)
Distributive Law. \(B \cap (\bigcup_{i\in I} A_i) = \bigcup_{i\in I}(B \cap A_i)\text{.}\)
Distributive Law. \(B \cup (\bigcap_{i\in I} A_i) = \bigcap_{i\in I} (B \cup A_i)\text{.}\)
De Morgan's Law. \(B \sm (\bigcup_{i\in I} A_i) = \bigcap_{i\in I} (B \sm A_i)\text{.}\)
De Morgan's Law. \(B \sm (\bigcap_{i\in I} A_i) = \bigcup_{i\in I} (B \sm A_i)\text{.}\)
