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 $j$.

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