Advanced mathematics. A transitional reference

Contents......Page 6
Preface......Page 8
Possible Beneficial Audiences......Page 10
Wow Factors of the Book......Page 11
Chapter by Chapter (the nitty-gritty)......Page 12
Note to the Reader......Page 14
About the Companion Website......Page 15
1 Logic and Proofs......Page 16
 1.1 Sentential Logic......Page 18
 1.2 Conditional and Biconditional Connectives......Page 39
 1.3 Predicate Logic......Page 53
 1.4 Mathematical Proofs......Page 66
 1.5 Proofs in Predicate Logic......Page 86
 1.6 Proof by Mathematical Induction......Page 98
2 Sets and Counting......Page 110
 2.1 Basic Operations of Sets......Page 112
 2.2 Families of Sets......Page 130
 2.3 Counting: The Art of Enumeration......Page 140
 2.4 Cardinality of Sets......Page 158
 2.5 Uncountable Sets......Page 171
 2.6 Larger Infinities and the ZFC Axioms......Page 182
3 Relations......Page 194
 3.1 Relations......Page 196
 3.2 Order Relations......Page 210
 3.3 Equivalence Relations......Page 227
 3.4 The Function Relation......Page 239
 3.5 Image of a Set......Page 257
4 The Real and Complex Number Systems......Page 270
 4.1 Construction of the Real Numbers......Page 272
 4.2 The Complete Ordered Field: The Real Numbers......Page 284
 4.3 Complex Numbers......Page 296
5 Topology......Page 314
 5.1 Introduction to Graph Theory......Page 316
 5.2 Directed Graphs......Page 336
 5.3 Geometric Topology......Page 349
 5.4 Point-Set Topology on the Real Line......Page 364
6 Algebra......Page 382
 6.1 Symmetries and Algebraic Systems......Page 384
 6.2 Introduction to the Algebraic Group......Page 400
 6.3 Permutation Groups......Page 418
 6.4 Subgroups: Groups Inside a Group......Page 434
 6.5 Rings and Fields......Page 448
Index......Page 458