I Sets, Numbers, and Graphs......Page 14
1 Fundamentals: Concepts and Logic......Page 16
1.1 Propositional Logic......Page 17
1.2 Architecture of Concepts......Page 21
2 Axiomatic Set Theory......Page 28
2.1 The Axioms......Page 30
2.2 Basic Concepts and Results......Page 33
3.1 The Boolean Algebra of Subsets......Page 38
4.1 Graphs and Functions......Page 42
4.2 Relations......Page 54
5.1 Ordinal Numbers......Page 58
5.2 Natural Numbers......Page 63
6 Recursion Theorem and Universal Properties......Page 68
6.1 Recursion Theorem......Page 69
6.2 Universal Properties......Page 71
6.3 Universal Properties in Relational Database Theory......Page 79
7.1 Natural Operations......Page 86
7.2 Euclid and the Normal Forms......Page 89
8.1 The Diagonalization Procedure......Page 92
9 The Classical Number Domains Z, Q, R, and C......Page 94
9.1 Integers Z......Page 95
9.2 Rationals Q......Page 100
9.3 Real Numbers R......Page 103
9.4 Complex Numbers C......Page 115
10 Categories of Graphs......Page 120
10.1 Directed and Undirected Graphs......Page 121
10.2 Morphisms of Digraphs and Graphs......Page 127
10.3 Cycles......Page 138
11 Construction of Graphs......Page 142
12.1 n-ary Trees......Page 150
12.2 Moore Graphs......Page 152
13.1 Euler's Formula for Polyhedra......Page 156
13.2 Kuratowski's Planarity Theorem......Page 160
14.1 Floating Point Arithmetic......Page 162
14.2 Example for an Addition......Page 167
II Algebra, Formal Logic, and Linear Geometry......Page 170
15.1 Monoids......Page 172
15.2 Groups......Page 176
15.3 Rings......Page 184
15.4 Fields......Page 190
16.1 Prime Factorization......Page 194
16.2 Roots of Polynomials and Interpolation......Page 199
17 Formal Propositional Logic......Page 204
17.1 Syntactics: The Language of Formal Propositional Logic......Page 206
17.2 Semantics: Logical Algebras......Page 209
17.3 Signification: Valuations......Page 213
17.4 Axiomatics......Page 216
18 Formal Predicate Logic......Page 222
18.1 Syntactics: First??order Language......Page 224
18.2 Semantics: Sigma??Structures......Page 230
18.3 Signification: Models......Page 231
19 Languages, Grammars, and Automata......Page 236
19.1 Languages......Page 237
19.2 Grammars......Page 242
19.3 Automata and Acceptors......Page 256
20 Categories of Matrixes......Page 274
20.1 What Matrixes Are......Page 275
20.2 Standard Operations on Matrixes......Page 278
20.3 Square Matrixes and their Determinant......Page 284
21 Modules and Vector Spaces......Page 292
22 Linear Dependence, Bases, and Dimension......Page 300
22.1 Bases in Vector Spaces......Page 301
22.2 Equations......Page 308
22.3 Affine Homomorphisms......Page 309
23.1 Gauss Elimination......Page 316
23.2 The LUP Decomposition......Page 320
24.1 Euclidean Vector Spaces......Page 324
24.2 Trigonometric Functions from Two??Dimensional Rotations......Page 333
24.3 Gram's Determinant and the Schwarz Inequality......Page 336
25.1 Eigenvalues and Rotations......Page 340
25.2 The Vector Product......Page 344
25.3 Quaternions......Page 346
26.1 Galois Fields......Page 356
26.2 The Reed??Solomon (RS) Error Correction Code......Page 362
26.3 The Rivest??Shamir??Adleman (RSA) Encryption Algorithm......Page 366
A Further Reading......Page 370
B Bibliography......Page 372
Index......Page 376