2.4. Boolean operators on sets
Union of a family of sets
For any family of unary predicates A_{i} with
index i ∈ I, 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
metaoperation between unary predicates, with result a unary
predicate ℛ, defined for x in C as Qi, A_{i}(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 ∈ ⋃
i∈I
F_{i} ⇔ ∃i∈I, x∈F_{i}
This class is a set independent of E, as it can also be
defined from the union of a set of sets (1.11):
⋃
i∈I
F_{i} = ⋃{F_{i}
 i ∈ I}
(∀x∈⋃
i∈I
F_{i}, B(x))
⇔ ∀i∈I, ∀x∈F_{i},
B(x)
x ∈ X∪Y
⇔ (x ∈ X ∨ x ∈ Y)
X ⊂ X∪Y = Y∪X
All extensional definition operators (except ∅) are definable from
pairing and binary union.
Intersection
Any family of sets (F_{i})_{i∈I}
can be seen as a family of subsets of any set E that includes their
union U = ⋃_{i∈I} F_{i} ⊂ E. Then
for any operation (quantifier) Q between Booleans indexed by
I, the predicate ℛ(x) defined as (Qi, x
∈ F_{i}) takes value (Qi, 0) for all x
∉ U. Thus Q needs to satisfy ¬(Qi,0) (to be
false when all entries are false) for the class ℛ to be a set
{x∈E ℛ(x)} = {x∈U ℛ(x)}. This
was the case for Q=∃, including on the empty family (⋃∅ =∅),
but for ∀ (that defines the intersection) it requires a nonempty
family of sets (I ≠ ∅):
∀j ∈ I, ∩
i∈I F_{i} =
{x∈F_{j}  ∀i∈I,
x∈F_{i}}
x ∈ ∩
i∈I
F_{i} ⇔ ∀i∈I, x
∈ F_{i}
x ∈ X⋂Y ⇔ (x∈X ∧ x∈Y)
X⋂Y = Y⋂X = {x∈X  x∈Y} ⊂ X
X = X∪Y ⇔ Y ⊂ X ⇔ Y
= X⋂Y
Two sets A and B are called disjoint when A⋂B=∅,
which is equivalent to ∀x∈A, x∉B.
Union and intersection have the same associativity and
distributivity properties as ∧ and ∨ : A∪B∪C = (A∪B)∪C
= A∪(B∪C) = ⋃{A,B,C}
(

⋃
i∈I 
A_{i})∩C
=

⋃
i∈I 
(A_{i}∩C) 
(

∩
i∈I 
A_{i})∪C
= 
∩
i∈I 
(A_{i}∪C) 
(A∪B)⋂C
= 
(A⋂C)∪(B⋂C)

(A⋂B)∪C
= 
(A∪C)⋂(B∪C)

Other operators
The difference is defined by A\B = {x∈A
x ∉ B} so that x ∈ A\B ⇔ x∈A
∧ x∉B.
Finally the connective ⇎ gives the symmetric difference: A∆B
= (A∪B)\(A∩B).
When (Qi,0) is true, we must choose a set E to define
operations between subsets of E:
∩
i∈I
F_{i} = ⋂{F_{i}
 i∈I} = {x∈E
 ∀i∈I, x ∈ F_{i}} =
∁_{E} ⋃
i∈I
∁_{E} F_{i}