Introduction to Mathematical Proofs. A Transition to Advanced Mathematics

Description

Contents......Page 3 Preface......Page 5 1 Logic......Page 8 1.1 Statements, Negation, and Compound Statements......Page 9 1.2 Truth Tables and Logical Equivalences......Page 14 1.3 Conditional and Biconditional Statements......Page 26 1.4 Logical Arguments......Page 32 1.5 Open Statements and Quantifiers......Page 45 1.6 Chapter Review......Page 58 2.1 Deductive Mathematical Systems......Page 67 2.2 Mathematical Proofs......Page 78 2.3 Chapter Review......Page 113 3 Set Theory......Page 122 3.1 Sets and Subsets......Page 123 3.2 Set Operations......Page 128 3.3 Additional Set Operations......Page 136 3.4 Generalized Set Union and Intersection......Page 148 3.5 Chapter Review......Page 155 4.1 Relations......Page 163 4.2 The Order Relations......Page 172 4.3 Reflexive, Symmetric, Transitive, and Equivalence Relations......Page 180 4.4 Equivalence Relations, Equivalence Classes, and Partitions......Page 186 4.5 Chapter Review......Page 193 5.1 Functions......Page 199 5.2 Onto Functions, One-to-One Functions, and One-toOne Correspondences......Page 209 5.3 Inverse of a Function......Page 216 5.4 Images and Inverse Images of Sets......Page 224 5.5 Chapter Review......Page 230 6.1 Mathematical Induction......Page 235 6.2 The Well-Ordering Principle and the Fundamental Theorem of Arithmetic......Page 240 7 Cardinalities of Sets......Page 250 7.1 Finite Sets......Page 251 7.2 Denumerable and Countable Sets......Page 257 7.3 Uncountable Sets......Page 262 8.1 Sequences......Page 271 8.2 Limit Theorems for Sequences......Page 279 8.3 Monotone Sequences and Subsequences......Page 284 8.4 Cauchy Sequences......Page 291 9.1 Binary Operations and Algebraic Structures......Page 294 9.2 Groups......Page 299 9.3 Subgroups and Cyclic Groups......Page 306 Reading Mathematical Proofs......Page 314 Writing Mathematical Proofs......Page 315 Answers......Page 321 Refs......Page 389