Advanced Mathematics : A Transitional Reference

Provides 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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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 mathematics courses. Topics include sentential and predicate calculus, mathematical induction, sets and counting, complex numbers, point-set topology, and symmetries, abstract groups, rings, and fields. Each section contains numerous problems for students of various interests and abilities. Ideally suited for a one-semester course, this book: Introduces students to mathematical proofs and rigorous thinking Provides thoroughly class-tested material from the authors own course in transitioning to higher math Strengthens the mathematical thought process of the reader Includes informative sidebars, historical notes, and plentiful graphics Offers a companion website to access a supplemental solutions manual for instructors Advanced Mathematics: A Transitional Reference is a valuable guide for undergraduate students who have taken courses in calculus, differential equations, or linear algebra, but may not be prepared for the more advanced courses of real analysis, abstract algebra, and number theory that await them. This text is also useful for scientists, engineers, and others seeking to refresh their skills in advanced math. 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 La 4ème de couverture indique : "Provides 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 mathematics courses. Topics include sentential and predicate calculus, mathematical induction, sets and counting, complex numbers, point-set topology, and symmetries, abstract groups, rings, and fields. Each section contains numerous problems for students of various interests and abilities. "