Set theory and Foundations of Mathematics 

1.
First
foundations of mathematics 

2.
Set theory (continued) (old pdf version : 11 pages) 

2.1. Tuples, families

Tuples Conditional connective Families Structures and binders Extensional definition of sets 
2.2. Boolean operators on families of sets  Union of a family of sets Intersection Other operators 
2.3. Products, graphs and composition  Finite product Sum or disjoint union Composition, restriction, graph of a function Direct image, inverse image 
2.4. Uniqueness quantifiers, functional graphs  Definitions of the uniqueness quantifiers Translating operators into predicates Conditional operator Functional graphs 
2.5. The powerset axiom  Need of additional primitive symbols Powerset Exponentiation Product Cantor Theorem 
2.6. Injectivity and inversion  Injections Inverse Properties of injectivity, surjectivity, composition and inversion 
2.7. Properties
of binary relations on a set ; ordered sets 
Properties of binary relations Preorder Order Strict order, total order Monotone and antitone functions Order on sets of functions 
2.8. Canonical bijections  Notion of canonical bijection Sum of sets, sum of functions Product of functions or recurrying 
2.9. Equivalence relations and partitions  Indexed partitions Equivalence relation defined by a function Partition, canonical surjection Order quotient of a preorder 
2.10. Axiom of choice  6 simply equivalent formulations 
2.11. Notion
of Galois connection (basic properties ; see text Galois connections for more developments) 
Fixed points; idempotence Notion of Galois connection General example Properties Closures 