reflexive ⇔  ∀x, x R x 
irreflexive ⇔  ∀x, ¬(x R x) 
symmetric ⇔  R ⊂ ^{t}R ⇔ R = ^{t}R ⇔ ⃗R = ⃖R 
antisymmetric ⇔  ∀x,y, (x R y∧y R x)⇒x=y 
transitive ⇔  ∀x,y,z, (x R y∧y R z)⇒x R z 
(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 without any pair of incomparable elements : ∀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 (continued)  
2.1. Tuples, families
2.2. Boolean operators on families of sets 2.3. Products, graphs and composition 2.4. Uniqueness quantifiers, functional graphs 2.5. The powerset axiom 2.6. Injectivity and inversion ⇦ 
2.7. Properties of binary relations on a
set ; ordered sets ⇨2.8. Canonical bijections 2.9. Equivalence relations and partitions 2.10. Axiom of choice 2.11. Notion of Galois connection 
