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Sets, Numbers, and Graphs
Fundamentals: Concepts and Logic
Propositional Logic
Architecture of Concepts
Axiomatic Set Theory
The Axioms
Basic Concepts and Results
The Boolean Algebra of Subsets
Graphs and Functions
Relations
Ordinal Numbers
Natural Numbers
Recursion Theorem and Universal Properties
Recursion Theorem
Universal Properties
Universal Properties in Relational Database Theory
Natural Operations
Euclid and the Normal Forms
The Diagonalization Procedure
The Classical Number Domains Z, Q, R, and C
Integers Z
Rationals Q
Real Numbers R
Complex Numbers C
Categories of Graphs
Directed and Undirected Graphs
Morphisms of Digraphs and Graphs
Cycles
Construction of Graphs
n-ary Trees
Moore Graphs
Euler's Formula for Polyhedra
Kuratowski's Planarity Theorem
Floating Point Arithmetic
Example for an Addition
Algebra, Formal Logic, and Linear Geometry
Monoids
Groups
Rings
Fields
Prime Factorization
Roots of Polynomials and Interpolation
Formal Propositional Logic
Syntactics: The Language of Formal Propositional Logic
Semantics: Logical Algebras
Signification: Valuations
Axiomatics
Formal Predicate Logic
Syntactics: First?order Language
Semantics: Sigma?Structures
Signification: Models
Languages, Grammars, and Automata
Languages
Grammars
Automata and Acceptors
Categories of Matrixes
What Matrixes Are
Standard Operations on Matrixes
Square Matrixes and their Determinant
Modules and Vector Spaces
Linear Dependence, Bases, and Dimension
Bases in Vector Spaces
Equations
Affine Homomorphisms
Gauss Elimination
The LUP Decomposition
Euclidean Vector Spaces
Trigonometric Functions from Two?Dimensional Rotations
Gram's Determinant and the Schwarz Inequality
Eigenvalues and Rotations
The Vector Product
Quaternions
Galois Fields
The Reed?Solomon (RS) Error Correction Code
The Rivest?Shamir?Adleman (RSA) Encryption Algorithm
A Further Reading
B Bibliography
Index