Table of Contents
Preface vii
Possible Beneficial Audiences ix
Wow Factors of the Book x
Chapter by Chapter (the nitty-gritty) xi
Note to the Reader xiii
About the Companion Website xiv
Chapter 1 Logic and Proofs 1
1.1 Sentential Logic 3
1.2 Conditional and Biconditional Connectives 24
1.3 Predicate Logic 38
1.4 Mathematical Proofs 51
1.5 Proofs in Predicate Logic 71
1.6 Proof by Mathematical Induction 83
Chapter 2 Sets and Counting 95
2.1 Basic Operations of Sets 97
2.2 Families of Sets 115
2.3 Counting: The Art of Enumeration 125
2.4 Cardinality of Sets 143
2.5 Uncountable Sets 156
2.6 Larger Infinities and the ZFC Axioms 167
Chapter 3 Relations 179
3.1 Relations 181
3.2 Order Relations 195
3.3 Equivalence Relations 212
3.4 The Function Relation 224
3.5 Image of a Set 242
Chapter 4 The Real and Complex Number Systems 255
4.1 Construction of the Real Numbers 257
4.2 The Complete Ordered Field: The Real Numbers 269
4.3 Complex Numbers 281
Chapter 5 Topology 299
5.1 Introduction to Graph Theory 301
5.2 Directed Graphs 321
5.3 Geometric Topology 334
5.4 Point-Set Topology on the Real Line 349
Chapter 6 Algebra 367
6.1 Symmetries and Algebraic Systems 369
6.2 Introduction to the Algebraic Group 385
6.3 Permutation Groups 403
6.4 Subgroups: Groups Inside a Group 419
6.5 Rings and Fields 433
Index 443