Description
Book SynopsisProvides a smooth and pleasant transition from first-year calculus to upper-level mathematics courses in real analysis, abstract algebra and number theory
Most universities require students majoring in mathematics to take a transition to higher math course that introduces mathematical proofs and more rigorous thinking. Such courses help students be prepared for higher-level mathematics course from their onset. Advanced Mathematics: A Transitional Reference provides a crash course in beginning pure mathematics, offering instruction on a blendof inductive and deductive reasoning. By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive setting.
Clear and concise chapters cover all the essential topics students need to transition from the rote-orientated courses of calculus to the more rigorous proof-orientated advanced mathematic
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
