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Needs:
Families
Set Unions and Intersections
Generalized Set Dualities
Needed by:
Family Products and Unions
Inverses Unions Intersections and Complements
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Family Unions and Intersections

Why

We can use families to think about unions and intersections.

Family unions

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 $\cup_{i \in I} A_i$.

$(x \in \cup_{i \in I} A_i) \iff (\exists i)(x \in A_i)$

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.

Let $\set{I_j}$ be a family of sets and define $K = \union_{j} I_j$. Then $\union_{k \in K} A_k = \union_{j \in J}(\union_{i \in I_j} A_i)$.2

Family intersection

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 $\cap _{i \in I} A_i$.

$x \in \cap _{i \in I} A_i \iff (\forall i)(x \in A_i)$

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).

Connections

The following are easy.4

Let $\set{A_i}$ be a family of subsets of $X$ and let $B \subset X$.

$B \cap \bigcup_{i} A_i = \bigcup_{i} (B \cap A_i)$
$B \cup \bigcap_{i} A_i = \bigcap_{i} (B \cup A_i)$

Let $\set{A_i}$ and $\set{B_j}$ be families of sets.5

$(\bigcup_{i} A_i) \cap (\bigcup_{j} B_j) = \bigcup_{i,j}(A_i \cap B_j)$
$(\bigcap_{i} A_u) \cup (\bigcap_{j} B_j) = \bigcap_{i,j}(A_i \cup B_j).$
$\cap _i X_i \subset X_j \subset \cup_i X_i$ for each $j$.

  1. The commutative law will appear in future editions. ↩︎
  2. An account will appear in future editions. ↩︎
  3. Statements of these will be given in future editions. ↩︎
  4. Nevertheless, full accounts will appear in future editions. ↩︎
  5. An account of the notation used and the proofs will appear in future editions. ↩︎
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