1.4 Indexed Collections of Sets
Given many sets: \[ A_1, A_2, \dots, A_n \quad n \text{ sets, } n \in \mathbb{N} \]
Union: \[ \begin{align} A_1 \cup A_2 \cup \dots \cup A_n &= \bigcup_{i=1}^n{A_i}\\ &= \{ x \in \mathbb{U}: x \in A_i \text{ for some } i, i=1\dots n \} \end{align} \]
Intersection: \[A_1 \cap A_2 \cap \dots \cap A_n = \bigcap_{i=1}^n{A_i}\]
Example:
For \(n \in \mathbb{N}, S_n = \{ n, 2n \}\). Suppose \(I = \{ 1, 2, 4 \}.\)
\[
\text{Find }\bigcup_{\alpha \in I}{S_\alpha}
\]
\[ \bigcup_{\alpha \in I}{S_\alpha} = \bigcup_{\alpha \in \{ 1, 2, 4 \}}{S_\alpha} = S_1 \cup S_2 \cup S_3 \] \[ = \{ 1, 2 \} \cup \{ 2, 4 \} \cup \{ 4, 8 \} = \{ 1, 2, 4, 8 \} \]