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Levi Pearson 2014-01-25 15:51:10 -07:00
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math/proof/notes.page
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@ -216,15 +216,28 @@ $$ so $\overline P$ is the set of composite numbers and 1.
**Definition 1.7** **Definition 1.7**
$A_1, A_2,\dots, A_n$ are sets. $A_1, A_2,\dots, A_n$ are sets. Then
$$\begin{align} $$\begin{align}
A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n &= A_1 \cup A_2 \cup A_3 \cup \dots \cup A_n &=
\left\{x : x \in A_i \left\{x : x \in A_i
\text{ for at least one set $A_i$, for } \text{ for at least one set $A_i$, for }
1 \leq i \leq n\right\}\\ 1 \leq i \leq n\right\}\\
A_1 \cap A_2 \cap A_3 \cap \dots \cap A-n &= A_1 \cap A_2 \cap A_3 \cap \dots \cap A_n &=
\left\{x : x \in A_i \left\{x : x \in A_i
\text{ for every set $A_i$, for } \text{ for every set $A_i$, for }
1 \leq i \leq n \right\}\\ 1 \leq i \leq n \right\}\\
\end{align}$$ \end{align}$$
Given $A_1, A_2, \dots, A_n$ we define
$$\begin{align}
\bigcup{i=1}{n} A_i &= A_1 \cup A_2 \cup \dots \cup A_n\\
\bigcap{i=1}{n} A_i &= A_1 \cap A_2 \cap \dots \cap A_n\\
\end{align}$$
$$
A_1 = \{-1,0,1}, A_2 = \{-2,0,2}, \dots, A_i = \{-i,0,i}, \dots\\
\bigcup{i=1}{\inf} A_i = \Bbb{Z} \text{ and }
\bigcap{i=1}{\inf} A_i = \{0\}\\
$$