2.4. Boolean operators on sets

Union of a family of sets

For any family of unary predicates Ai with index iI, all definite in (at least) a common class C, their application to the same variable x with range C reduces this to a family of Boolean variables depending on x (their parameter, see 1.5). This way, any operation Q on these variables (a quantifier with range I, which becomes a connective when I is finite) defines a meta-operation between unary predicates, with result a unary predicate ℛ, defined for x in C as Qi, Ai(x). When C is a set E, this operates between subsets of E (through ∈ and the set builder).

For example, the existential quantifier (Q = ∃) defines the union of a family of sets:

x
iI
Fi ⇔ ∃iI, xFi
This class is a set independent of E, as it can also be defined from the union of a set of sets (1.11):

iI
Fi ={Fi | iI}
(x
iI
Fi, B(x)) ⇔ ∀iI, ∀xFi, B(x)
xXY ⇔ (xXxY)
XXY = YX

All extensional definition operators (except ∅) are definable from pairing and binary union.

Intersection

Any family of sets (Fi)iI can be seen as a family of subsets of any set E that includes their union U = ⋃iI FiE. Then for any operation (quantifier) Q between Booleans indexed by I, the predicate ℛ(x) defined as (Qi, xFi) takes value (Qi, 0) for all xU. Thus Q needs to satisfy ¬(Qi,0) (to be false when all entries are false) for the class ℛ to be a set {xE| ℛ(x)} = {xU| ℛ(x)}.

This was the case for Q=∃, including on the empty family (⋃∅ =∅), but for ∀ (that defines the intersection) it requires a non-empty family of sets (I ≠ ∅):

 ∀jI,
iI
Fi = {xFj | ∀iI, xFi}
  x
iI
Fi ⇔ ∀iI, xFi
xXY ⇔ (xXxY)
XY = YX = {xX | xY} ⊂ X
X = XYYXY = XY

Two sets A and B are called disjoint when AB=∅, which is equivalent to ∀xA, xB.
Union and intersection have the same associativity and distributivity properties as ∧ and ∨ :

ABC = (AB)∪C = A∪(BC) ={A,B,C}

(
iI
Ai)C =

iI
(AiC) (
iI
Ai)C =
iI
(AiC)
(AB)⋂C =  (AC)∪(BC) (AB)∪C =  (AC)⋂(BC)

Other operators

The difference is defined by A\B = {xA| xB} so that xA\B ⇔ xAxB.
Finally the connective ⇎ gives the symmetric difference: AB = (AB)\(AB).
When (Qi,0) is true, we must choose a set E to define operations between subsets of E:

iI
Fi = {Fi | iI} = {xE | ∀iI, xFi} = ∁E
iI
E Fi

Set theory and Foundations of Mathematics
1. First foundations of mathematics
2. Set theory
2.1. Formalization of set theory
2.2. Set generation principle
2.3. Tuples, families
2.4. Boolean operators on sets
2.5. Products, graphs and composition
2.6. Uniqueness quantifiers, functional graphs
2.7. The powerset
2.8. Injectivity and inversion
2.9. Binary relations ; order
2.10. Canonical bijections
2.11. Equivalence relations, partitions
2.12. Axiom of choice
2.13. Galois connection
Time in set theory
Interpretation of classes
Concepts of truth in mathematics
3. Algebra - 4. Arithmetic - 5. Second-order foundations