CSE 260 Fall 2005

Mathematical Foundations of Computer Science

Welcome Page

# The Mathematics Genealogy Project

CSE 260 Course Topics:

• Mathematical reasoning and proof principles, Logic
• Set Theory
• Relations, Functions, Partial Functions
• Induction on N
• Composition of relations and functions
• Recursion on N
• Injection, surjections, bijections
• Inverses of functions and relations
• Direct and Inverse Images
• Equinumerosity, Cantor's Theorem,
• Pigeonhole Principle, Schroeder-Bernstein's Theorem
• Strings, Multisets, Indexed Families
• An amazing surjection: Hilbert's square-filling curve
• Some counting problems
• Binomial coefficients, the binomial formula
• Partial Orders, Lattices
• Well-Founded Orderings and complete Induction
• Equivalence Relations and Partitions. Closures.
• Distributive Lattices, Boolean algebras, Heyting algebras
• Trees, Multiset Ordering, String and Tree Embedding
• Graphs (directed and undirected; unlabeled and labeled); Graph homomorphisms and isomorphisms
• Simple Combinatorics--Counting
• Baby Number Theory: divisibility, modular arithmetic

# Related Material: Some Talks, Slides and Links

• Intuitionistic logic for dummies (pdf)
• Intuitionistic logic (Part 2/2) (pdf)
• Introduction to proof theory (pdf)
• Proofs are programs (ppt)
• Intuitionistic Logic (from Stanford Encyclopedia of Philosophy) (html)
• Intuitionistic Logic (From Wikipedia) (html)
• Intuitionistic Logic (From Mathworld) (html)
• Intuitionistic Logic (From Absolute Astronomy) (html)
• Axioms of Zermelo-Fraenkel Set Theory (from Wikipedia) (html)
• Zermelo-Fraenkel Axioms (from MathWorld) (html)
• Zermelo-Fraenkel Set Theory (from Stanford Encyclopedia of Philosophy) (html)
• Set Theory (from Stanford Encyclopedia of Philosophy) (html)
• The Axiom of Choice (by Eric Schechter) (html)
• Axiom of choice (from Wikipedia) (html)
• Axiom of choice (from MathWorld) (html)
• Cantor-Schroeder-Bernstein Theorem (from Wikipedia) (html)
• Cantor-Schroeder-Bernstein Theorem (from PlanetMath) (html)
• Tarski's Fixed Point Theorem (from MathWorld) (html)
• Knaster-Tarski Theorem (from Wikipedia) (html)
• Plane-Filling Function (from MathWorld) (html)
• Hilbert Curve (from MathWorld) (html)
• Plane Filling Curves (from Alexander Bogomolny) (html)
• Space-Filling Curves (from Wikipedia) (html)
• Using Space-Filling Curves for multi-dimensional indexing (html)
• Stirling's Formula (from Wikipedia) (html)
• Well-founded relation (from Wikipedia) (html)
• Well-founded induction (from PlanetMath) (html)
• Lattice (from Wikipedia) (html)
• Distributive Lattice (from Wikipedia) (html)
• Complete Lattice (from Wikipedia) (html)
• Boolean algebra (from Wikipedia) (html)
• Boolean algebra (from Stanford Encyclopedia of Philosophy) (html)
• Graph theory (from Wikipedia) (html)

# Some Course Notes and Slides

• Discrete math. basics, induction, inductive definitions
(Chapter from ``Logic for Computer Science'', by J. Gallier)   (ps)   |  (pdf)
• Macro prooftree.tex (tex)
• Macro mac.tex (tex)
• Macro mathmac.tex (tex)
• Discrete Mathematics for Computer Science (book manuscript)   (html)