More notes

math/proof/notes.page

@@ -236,8 +236,27 @@ $$\begin{align}
 \bigcap_{i=1}^{n} A_i &= A_1 \cap A_2 \cap \dots \cap A_n\\
 \end{align}$$
 
+For example, take the following infinite list of sets:
+
+$$
+A_1 = \{-1,0,1\}, A_2 = \{-2,0,2\}, \dots, A_i = \{-i,0,i\}, \dots
+$$
+
+We can see that:
+
 $$
-A_1 = \{-1,0,1\}, A_2 = \{-2,0,2\}, \dots, A_i = \{-i,0,i\}, \dots\\
 \bigcup_{i=1}^{\inf} A_i = \Bbb{Z} \text{ and }
 \bigcap_{i=1}^{\inf} A_i = \{0\}\\
 $$
+
+**Definition 1.8**
+
+If we have a set $A_{\alpha}$ for every $\alpha$ in some index set $I$,
+then
+
+$$\begin{align}
+\bigcup_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text {
+for at least one set $A_{\alpha}$ with } \alpha \in I \}\\
+\bigcap_{\alpha \in I} A_{\alpha} &= \{ x : x \in A_{\alpha} \text {
+for every set $A_{\alpha}$ with } \alpha \in I \}\\
+\end{align}$$