reflexive ⇔  ∀x, xRx 
irreflexive ⇔  ∀x, ¬(xRx) 
symmetric ⇔  R ⊂ ^{t}R ⇔ R = ^{t}R ⇔ ⃗R = ⃖R 
antisymmetric ⇔  ∀x,y, (xRy ∧ yRx) ⇒ x=y 
transitive ⇔  ∀x,y,z, (xRy ∧ yRz) ⇒ xRz 
(Id_{E} ∈ A) ⇒  R is reflexive 
(∀f,g ∈ A, g০f ∈ A) ⇒  R is transitive 
(∀f∈A, f:E↔ E ∧ f^{−1}∈A) ⇒  R is symmetric 
Then ^{t}R is also a preorder: x R y ⇔ ⃗R(y) ⊂ ⃗R(x).
Ordered set. An order is an antisymmetric preorder. A preordered set is a set E with a chosen preorder R. An ordered set is a set with an order, usually written as ≤ .For x,y in an ordered set E, the formula x ≤ y can be read «x is less than y», or «y is greater than x». The elements x and y are said incomparable when ¬(x≤y ∨ y≤x). (This implies x≠y).
Any subset F of a set E with an order (resp. a preorder) R ⊂ E×E, is also ordered (resp. preordered) by its restriction R∩(F×F) (which is an order, resp. preorder, on F).Strict order. It is a binary relation both
transitive and irreflexive; and thus also antisymmetric.
Total order. A total order on a set E is an order R on E where all elements are comparable : ∀x,y∈E, x≤y ∨ y≤x, i.e. R∪^{t}R = E×E.
Equivalent definitions:f ≤ g ⇔ (∀x∈E, f(x) ≤ g(x))
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 families of 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. Secondorder foundations 