Relations, Functions & Cardinality
A 22-module path through relations, functions & cardinality, in study order from foundations to the frontier.
Modules
- Ordered Pairs and Cartesian Products
- Relations as Sets of Ordered Pairs
- Properties of Relations: Reflexive, Symmetric, Transitive
- Equivalence Relations
- Equivalence Classes and Partitions
- Quotient Sets and Well-Definedness
- Partial Orders
- Total Orders and Chains
- Well-Orders and Least Elements
- Functions as Special Relations
- Injective Functions
- Surjective Functions
- Bijections and Inverse Functions
- Composition of Functions
- Images and Preimages of Sets
- Finite Sets and Counting
- Comparing Cardinalities
- Countable Sets
- Uncountable Sets and Cantor's Diagonal Argument
- The Schroeder-Bernstein Theorem
- Cardinal Arithmetic and the Power Set
- The Continuum Hypothesis