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Advanced Mathematics: A Transitional Reference





Advanced Mathematics: A Transitional Reference

Stanley J. Farlow

ISBN: 978-1-119-56351-8 October 2019 480 Pages



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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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. 


Possible Beneficial Audiences

Wow Factors of the Book

Chapter by Chapter (the nitty gritty)

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