Finite operations are useful, but only so far. Of course, restricting attention to finite operations may be useful when you want to do computation, but often it is more useful to use more general operations to prove theorems, and only later come back to find an efficient way to compute.
A set whose elements are themselves sets is often called a family or collection of sets. Technically, we could just call this a set, but we often use one of these terms in order to aid our intuition and memory. Generally, we will use script letters, such as \(\mathcal{A,B,C},\ldots\) to denote families of sets. Again, this is not necessary, merely a helpful practice.
If \(\mathcal{A}\) is a family of sets, the union over \(\mathcal{A}\) is the set whose elements are in at least one of the sets in \(\mathcal{A}\text{.}\) Logically, this means
\begin{equation*} x \in \bigcup_{A \in \mathcal{A}} A \iff (\exists A \in \mathcal{A})(x \in A). \end{equation*}If \(\mathcal{A}\) is a family of sets, the intersection over \(\mathcal{A}\) is the set whose elements are in all of the sets in \(\mathcal{A}\text{.}\) Logically, this means
\begin{equation*} x \in \bigcap_{A \in \mathcal{A}} A \iff (\forall A \in \mathcal{A})(x \in A). \end{equation*}Note that if \(B \in \mathcal{A}\text{,}\) then
\begin{equation*} \bigcap_{A \in \mathcal{A}} A \subseteq B \subseteq \bigcup_{A \in \mathcal{A}} A. \end{equation*}Let \(\mathcal{A}\) be the collection \(\{A_n \mid n \in \mathbb{N} \}\) where \(A_n := \{1,2,\ldots, n\}\text{.}\) Then
\begin{equation*} \bigcap_{A \in \mathcal{A}} A = \{1\} \qquad \bigcup_{A \in \mathcal{A}} A = \mathbb{N}. \end{equation*}In the above example, notice how we used \(\mathbb{N}\) as a tool to specify our collection \(\mathcal{A}\) using set builder notation. This is a common phenomenon and we give it a name. We call a set an index set if it plays a role similar to that of \(\mathbb{N}\) in the previous example. Moreover, we say that it indexes the family \(\mathcal{A}\text{,}\) and we call \(\mathcal{A}\) an indexed family. This is because we can access (or lookup) any set in the family \(\mathcal{A}\) using an element of the index set. We call an elemenet of the index set an index (the plural of this is indices).
With this concept, we can rewrite the union and intersection with slightly different notation, which is often more useful:
\begin{equation*} \bigcup_{n \in \mathbb{N}} A_n := \bigcup_{A \in \mathcal{A}} A \qquad \bigcap_{n \in \mathbb{N}} A_n := \bigcap_{A \in \mathcal{A}} A \end{equation*}In fact, when the index set is \(\mathbb{N}\) or some contiguous string of integers, we often write the union and intersection with notation similar to summation notation from Calculus. That is,
\begin{equation*} \bigcup_{n=1}^{\infty} A_n := \bigcup_{n \in \mathbb{N}} A_n, \end{equation*}and similarly,
\begin{equation*} \bigcup_{n=1}^{k} A_n := A_1 \cup A_2 \cup \cdots \cup A_k. \end{equation*}And similarly for intersections.
An indexed family \(\mathcal{A} = \{ A_{\alpha} \mid \alpha \in I \}\) of sets is said to be pairwise disjoint if for any \(\alpha, \beta \in I\) with \(\alpha \not= \beta\text{,}\) \(A_{\alpha} \cap A_{\beta} = \emptyset\text{,}\) i.e., sets corresponding to different indices are disjoint.