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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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.
Sprache
Verlagsort
Verlagsgruppe
Zielgruppe
Maße
Höhe: 231 mm
Breite: 147 mm
Dicke: 28 mm
Gewicht
ISBN-13
978-1-119-56351-8 (9781119563518)
Schweitzer Klassifikation
Stanley J. Farlow, PhD, is Professor Emeritus of Mathematics, University of Maine, USA. He was a Professor of Mathematics at the University of Maine for 47 years from 1968 to 2016, doing research in control theory, PDEs, and neural networks (GMDH algorithm) as well as teaching graduate and undergraduate courses in real and complex analysis, topology, differential equations, statistics, and a transition to higher math course.
Autor*in
University of Maine
Preface
Possible Beneficial Audiences
Wow Factors of the Book
Chapter by Chapter (the nitty gritty)
Note to the Reader
Note to the Reader
Chapter 1 Logic and Proofs
Section 1.1 Sentential Logic
Section 1.2 Conditional and Biconditional Connectives
Section 1.3 Predicate Logic
Section 1.4 Mathematical Proofs
Section 1.5 Proofs in Predicate Logic
Section 1.6 Proof by Mathematical Induction
Chapter 2 Sets and Counting
Section 2.1 Basic Operations of Sets
Section 2.2 Families of Sets
Section 2.3 Counting: The Art of Enumeration
Section 2.4 Cardinality of Sets
Section 2.5 Uncountable Sets
Section 2.6 Larger Infinities and the ZFC Axioms
Chapter 3 Relations
Section 3.1 Relations
Section 3.2 Order Relations
Section 3.3 Equivalence Relations
Section 3.4 The Function Relation
Section 3.5 Image of a Set
Chapter 4 The Real and Complex Number Systems
Section 4.1 Construction of the Real Numbers
Section 4.2 The Complete Ordered Field: The Real Numbers
Section 4.3 Complex Numbers
Chapter 5 Topology
Section 5.1 Introduction to Graph Theory
Section 5.2 Directed Graphs
Section 5.3 Geometric Topology
Section 5.4 Point-Set Topology on the Real Line
Chapter 6 Algebra
Section 6.1 Symmetries and Algebraic Systems
Section 6.2 Introduction to the Algebraic Group
Section 6.3 Permutation Groups
Section 6.4 Subgroups: Groups Inside a Group
Section 6.5 Rings and Fields
Index
