Set theory is fundamental to all of mathematics. Almost every other mathematical concept (including the concept of numbers and functions) can be formalized in terms of sets. The material below includes basic and intermediate topics in set theory and contains typical problems with detailed solutions.
- Logic and Set Notation
- Introduction to Sets
- Set Operations and Venn Diagrams
- Set Identities
- Cartesian Product of Sets
- Binary Relations
- Properties of Relations
- Operations on Relations
- Composition of Relations
- Closures of Relations
- Equivalence Relations
- Equivalence Classes and Partitions
- Partial Orders
- Total Orders
- Hasse Diagrams
- Special Elements of Partially Ordered Sets
- Well Orders
- Lexicographic Orders
- Lattices
- Topological Sorting
- Well Ordering Principle
- Functions as Relations
- Injection, Surjection, Bijection
- Inverse Functions
- Composition of Functions
- Floor and Ceiling Functions
- Pigeonhole Principle
- Cardinality of a Set
- Countable and Uncountable Sets
- Comparing Cardinalities
- Cantor-Schröder-Bernstein Theorem
- Cantor’s Theorem
- Ordinal Numbers