A Transition to Advanced Mathematics

Preface Preface to the Student Chapter 1 Logic and Proofs 1 1.1 Propositions and Connectives 1 1.2 Conditionals and Biconditionals 9 1.3 Quantifiers 18 1.4 Basic Proof Methods I 27 1.5 Basic Proof Methods II 40 1.6 Proofs Involving Quantifiers 48 1.7 Additional Examples of Proofs 60 Chapter 2 Set Theory 71 2.1 Basic Concepts of Set Theory 71 2.2 Set Operations 79 2.3 Extended Set Operations and Indexed Families of Sets 89 2.4 Mathematical Induction 100 2.5 Equivalent Forms of Induction 114 2.6 Principles of Counting 122 Chapter 3 Relations and Partitions 135 3.1 Cartesian Products and Relations 135 3.2 Equivalence Relations 147 3.3 Partitions 157 3.4 Ordering Relations 163 3.5 Graphs 174 Chapter 4 Functions 185 4.1 Functions as Relations 185 4.2 Constructions of Functions 195 4.3 Functions That Are Onto; One-to-One Functions 205 4.4 One-to-One Correspondences and Inverse Functions 213 4.5 Images of Sets 220 4.6 Sequences 225 Chapter 5 Cardinality 233 5.1 Equivalent Sets; Finite Sets 233 5.2 Infinite Sets 242 5.3 Countable Sets 251 5.4 The Ordering of Cardinal Numbers 259 5.5 Comparability of Cardinal Numbers and the Axiom of Choice 267 Chapter 6 Concepts of Algebra 275 6.1 Algebraic Structures 275 6.2 Groups 283 6.3 Subgroups 292 6.4 Operation Preserving Maps 298 6.5 Rings and Fields 306 Chapter 7 Concepts of Analysis 315 7.1 Completeness of the Real Numbers 316 7.2 The Heine-Borel Theorem 324 7.3 The Bolzano-Weierstrass Theorem 336 7.4 The Bounded Monotone Sequence Theorem 341 7.5 Equivalents of Completeness 347 Answers to Selected Exercises 353 Index 393.