We can use families to think about unions and intersections.
Let $A: I \to \powerset{X}$ be a family of subsets. We refer to the union (see Set Unions) of the range (see Relations) of the family the family union. We denote it $\cup_{i \in I} A_i$.
If $I = \set{a, b}$ is a pair with $a \neq b$, then $\cup_{i \in I} = A_a \cup A_b$.
There is no loss of generality in considering family unions. Every set of sets is a family: consider the identity function from the set of sets to itself.
We can also show generalized associative and commutative law1 for unions.
If we have a nonempty family of subsets $A: I \to \powerset{X}$, we call the intersection (see Set Intersections) of the range of the family the family intersection. We denote it $\cap _{i \in I} A_i$.
Similarly we can derive associative and commutative laws for intersection.3 They can be derived as for unions, or from the facts of unions using generalized DeMorgan's laws (see Generalized Set Dualities).
The following are easy.4
Let $\set{A_i}$ be a family of subsets of $X$ and let $B \subset X$.
Let $\set{A_i}$ and $\set{B_j}$ be families of sets.5