Principles of Mathematics

Preface xi 1 Set Theory 1 1.1 Introduction, 1 1.2 Set Theory - Definitions, Notation, and Terminology - What is a Set?, 3 1.3 Sets Given by a Defining Property, 15 1.4 The Algebra of Sets, 25 1.5 The Power Set, 41 1.6 The Cartesian Product, 44 1.7 The Sets N, Z, and Q, 46 1.8 The Set R - Real Numbers I, 71 1.9 A Short Musing on Transfinite Arithmetic, 80 1.10 The Set R - Real Numbers II, 102 1.11 Supplementary Problems, 109 2 Logic 115 2.1 Introduction, 116 2.2 Propositional Calculus, 121 2.3 Arguments I, 146 2.4 Arguments II, 167 2.5 A Short Revisit to Set Theory, 171 2.6 Boolean Algebra, 173 2.7 Supplementary Problems, 177 3 Proofs 183 3.1 Introduction, 183 3.2 Direct Proof, 193 3.3 Indirect Proof, 212 3.4 Mathematical Induction, 218 3.5 Supplementary Problems, 241 4 Functions 247 4.1 Introduction, 247 4.2 Relations, 248 4.3 Functions, 274 4.4 Supplementary Problems, 321 5 Group Theory 327 5.1 Introduction, 327 5.2 Fundamental Concepts of Group Theory, 328 5.3 Subgroups, 356 5.4 Cyclic Groups, 382 5.5 Homomorphisms and Isomorphisms, 385 5.6 Normal Subgroups, 404 5.7 Centralizer, Normalizer, Stabilizer, 412 5.8 Quotient Group, 419 5.9 The Isomorphism Theorems, 427 5.10 Direct Product of Groups, 437 5.11 Supplementary Problems, 441 6 Linear Algebra 447 6.1 Introduction, 447 6.2 Vector Space, 449 6.3 Linear Dependence and Independence, 456 6.4 Basis and Dimension of a Vector Space, 461 6.5 Subspaces, 469 6.6 Linear Transformations - Linear Operators, 477 6.7 Isomorphism of Linear Spaces, 489 6.8 Linear Transformations and Matrices, 501 6.9 Linear Space Mmn, 507 6.10 Matrix Multiplication, 509 6.11 Some More Special Matrices. General Linear Group, 514 6.12 Rank of a Matrix, 525 6.13 Determinants, 534 6.14 The Inverse and the Rank of a Matrix Revisited, 541 6.15 More on Linear Operators, 547 6.16 Systems of Linear Equations I, 585 6.17 Systems of Linear Equations II, 600 6.18 The Basics of Eigenvalue and Eigenvector Theory, 613 6.19 Supplementary Problems, 635 Index 645