gr R ⇔ (Set R ∧ ∀x∈R, Fnc x ∧ Dom x = V_{2}).
For any binder Q and any graph G, the formula Qz∈G, A(z_{0},z_{1}) that binds the variable z = (z_{0}, z_{1}) on a binary structure A definite on G, can be seen as binding 2 variables z_{0}, z_{1} on A(z_{0}, z_{1}), and thus be denoted with a pair of variables: Q(x,y)∈G, A(x,y).
The transpose of an ordered pair is^{t}(x,y) = (y,x)
The transpose of a graph R is the image of transposition over it:^{t}R = {(y,x)(x,y) ∈ R}
We define the domain and the image of a graph as the respective images of π_{0} and π_{1} over it:Dom R = {x(x,y) ∈ R}
Im R = {y(x,y) ∈ R} = Dom ^{t}R
∀x, R⃗(x) = {y
 (z,y)∈R ∧ z=x}.
∀y, R⃖(y) = {x  (x,z)∈R ∧
z=y} = ^{t}R⃗ (y).
∀x,y,
y ∈ R⃗ (x) ⇔ (x,y)
∈ R ⇔ x ∈ R⃖
(y)
∀x, x ∈ Dom R ⇔ R⃗
(x) ≠ ∅
Dom R ⊂ E ⇒ Im R = ⋃_{x∈E} R⃗(x) ∧
∀y, R⃖(y) = {x∈E  (x,y) ∈ R}
R⃗_{E} =
(E∋x ↦ R⃗(x))
R⃖_{F} = (F∋y ↦
R⃖(y))
(R⃗) = R⃗_{Dom R}
(R⃖) = R⃖_{Im R}.
Gr f = {(x,f(x))  x ∈ Dom f}
∀x,y, (x,y) ∈ Gr f ⇔ (x ∈ Dom f ∧ y =
f(x))
Dom f = Dom Gr f
Im f = Im Gr f
Gr f ⊂ R  ⇔ ∀x∈Dom f, f(x) ∈ R⃗(x) 
R ⊂ Gr f 
⇔ (Dom R ⊂ Dom f ∧ ∀(x,y)∈R, y = f(x)) 
R = Gr f 
⇔ (Dom R ⊂ Dom f ∧ ∀x∈Dom f, R⃗(x) = {f(x)} ) 
∀x∈ Dom R, !:
R⃗(x)
∀x,y∈R, x_{0} = y_{0} ⇒
x_{1} = y_{1}.
For any set E we shall denote 𝛿_{E} = Gr Id_{E}.
(∀y≠z∈F, ¬∃x∈R⃖(y), x ∈ R⃖(z)) ⇔ (∀(x,y)∈R, ∀z∈F, xRz ⇒ y = z)
For any function f and any y we define the fiber of y under f asf_{•}(y) = {x∈Dom f  f(x) = y} = (Gr⃖ f)(y)
When f : E → F this defines a family f_{•F} = (f_{•}(y))_{y∈F} of pairwise disjoint subsets of E:∀y,z∈F, f_{•}(y) ∩ f_{•}(z)
≠ ∅ ⇒ ∃x∈ f_{•}(y) ∩ f_{•}(z),
y = f(x) = z
⋃_{y∈F} f_{•}(y) = E
Im f = {y∈F  f_{•}(y) ≠ ∅}.
R_{A} = {(x,y)∈R  x∈A} = ∐_{x∈A} R⃗(x)
The direct image of a set A by a graph R isR_{⋆}(B) = ^{t}R^{⋆}(B) = ⋃_{y∈B} R⃖(y) = {x  (x,y)∈R∧ y ∈ B} ⊂ Dom R.
For any family (B_{i})_{i ∈ I} of subsets of F, f_{⋆}(⋂_{i∈I }B_{i}) = ⋂_{i∈I} f_{⋆}(B_{i}) where intersections are respectively interpreted as subsets of F and E.
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 
2.A. Time in set theory 2.B. Interpretation of classes 2.C. Concepts of truth in mathematics 

3. Algebra  4. Arithmetic  5. Secondorder foundations 