Families use the formalism of functions disguised in the style of the symbols for tuples (themselves inapplicable due to their finiteness for infinite domains). The evaluator for a family u at i, instead of u(i), is written π_{i}(u) or u_{i} (looking like a metavariable symbol of variable). The family definer on a term t is written (t(i))_{i∈I} or sometimes with ellipsis as (t(0),…,t(n−1)) abbreviating the ntuple definer, instead of I ∋ i ↦ t(i). The argument i is called index, and the family is said indexed by its domain I. We may write u = (u_{i})_{i∈I} as a shorthand for Fnc u ∧ Dom u = I.
When formalizing model theory in set theory, the "lists" of components forming the contents of theories are basically families. More precisely:(B_{0}∧…∧B_{n−1}) ⇔
(∀i∈V_{n}, B_{i})
(B_{0}∨…∨B_{n−1}) ⇔
(∃i∈V_{n}, B_{i})
(C ∧ ∃x∈E, R(x))
⇔ (∃x∈E, C ∧ R(x))
(C ∨ ∀x∈E, R(x))
⇔ (∀x∈E, C ∨ R(x))
(C ⇒ ∀x∈E, R(x)) ⇔ (∀x∈E,
C ⇒ R(x))
((∃x∈E, R(x)) ⇒ C) ⇔
(∀x∈E, R(x) ⇒ C)
(∃x∈E, C) ⇔ (C ∧ E≠∅) ⇒
C ⇒ (C ∨ E=∅) ⇔ (∀x∈E, C)
{a,b,…} = Im(a,b,…).
The definition of V_{n} from 2.3 can be written {0,…,n−1}. Images of tuples are finite sets.{T(x) x∈E} = {T(x)}_{x∈E} = Im(E ∋ x ↦ T(x))
As this notation looks similar to the set builder, both may be combined into a third notation :{T(x) x∈E ∧ R(x)} = {T(x) x∈{y∈ER(y)}}
A function f is said constant when !: Im f. The constancy of a tuple is the chain of equalities:x=y=z ⇔ !:{x,y,z} ⇔ ((x=y)∧(y=z)) ⇒ x=z
The binder of union of any family of sets (F = (F_{i})_{i∈I} ∧ ∀i∈I, Set F_{i}), is defined from the previous functor of union of a set of sets (2.2) :
∀_{C }x, R(x) ⇔ Qi, A_{i}(x).
When C is a set E, this defines a genuine binder (resp. operation) acting in the class of subsets of E. However, the result depends a priori not only on the given family of sets but also on the chosen set E. This parameter may be kept implicit as fixed in some contexts, but in others it needs to be explicitly added as argument.
The functor so defined by the negation connective (¬) is the complement ∁_{E} (with set parameter E) : ∁_{E} F is called the complement of F in E. It is synonymous with the following binary operator \ of difference between any two sets, except that ∁_{E} F is considered only definite if F⊂E :∀_{Set }E,F, E\F = {x∈E 
x ∉ F}
∀_{Set }E,F, F⊂E ⇒ ∁_{E} F = E\F ⊂ E
∴ (∀x∈E, x ∈ ∁_{E} F ⇔ ¬x∈F)
In any of these, the existential quantifier (Q = ∃) defines the union binder:
X = {x∈U R(x)}.
Noting that∀x, x ∈ U ∨ (R(x) ⇔ Qi, 0)
the needed independence condition appears to be ¬Qi, 0 which also implies that X coincides with the class R.∀_{Set }E,F, ∀x, x∈ E\F ⇔ (x∈E ∧ x∉F).
The quantifier ∀ defines the intersection of any family of subsets F_{i} of a fixed set E:
The condition of independence from E is that I ≠ ∅. As above defined, the intersection
of the empty family gives E. The intersection of any nonempty family of sets (I ≠ ∅)
can be written
E∩F∩G = (E∩F)∩G
= E∩(F∩G) = ⋂(E,F,G)
E ∩  ⋃ i∈I 
F_{i} = 
⋃ i∈I 
E ∩ F_{i}  E ∪  ⋂ i∈I 
F_{i} =  ⋂ i∈I 
E ∪ F_{i} 
E∩(F∪G) =  (E∩F)∪(E∩G)  E∪(F∩G) =  (E∪F)∩(E∪G) 
Finally the connective ⇎ gives the symmetric difference: E ∆ F = (E∪F) \ (E∩F).
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 2.4. Uniqueness quantifiers ⇦ 2.5. Families, Boolean operators on sets ⇨ 2.6. Graphs 2.7. Products and powerset 
2.8. Injections, bijections 2.9. Properties of binary relations 2.10. Axiom of choice 
Time
in set theory Interpretation of classes Concepts of truth in mathematics 

3. Algebra  4. Arithmetic  5. Secondorder foundations 