Family Unions and Intersections

Why

We can use families to think about unions and intersections.

Family unions

Let $A : I \to P (X)$ 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}) ⟺ (\exists i) (x \in A_{i})

If $I = {a, b}$ 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

{I_{j}}

be a family of sets and define

K = \cup_{j} I_{j}

. Then

\cup_{k \in K} A_{k} = \cup_{j \in J} (\cup_{i \in I_{j}} A_{i})

.²

Family intersection

If we have a nonempty family of subsets $A : I \to P (X)$ , 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} ⟺ (\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 ${A_{i}}$ be a family of subsets of $X$ and let $B \subset X$ .

B \cap ⋃_{i} A_{i} = ⋃_{i} (B \cap A_{i})

B \cup ⋂_{i} A_{i} = ⋂_{i} (B \cup A_{i})

Let ${A_{i}}$ and ${B_{j}}$ be families of sets.⁵

(⋃_{i} A_{i}) \cap (⋃_{j} B_{j}) = ⋃_{i, j} (A_{i} \cap B_{j})

(⋂_{i} A_{u}) \cup (⋂_{j} B_{j}) = ⋂_{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. ↩︎