Book description
Provides a smooth and pleasant transition from firstyear calculus to upperlevel 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 higherlevel 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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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 "roteorientated" courses of calculus to the more rigorous "prooforientated” advanced mathematics courses. Topics include sentential and predicate calculus, mathematical induction, sets and counting, complex numbers, pointset topology, and symmetries, abstract groups, rings, and fields. Each section contains numerous problems for students of various interests and abilities. Ideally suited for a onesemester course, this book:
 Introduces students to mathematical proofs and rigorous thinking
 Provides thoroughly classtested 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.
Table of contents
 Cover
 Preface
 Possible Beneficial Audiences
 Wow Factors of the Book
 Chapter by Chapter (the nitty‐gritty)
 Note to the Reader
 About the Companion Website

Chapter 1: Logic and Proofs

1.1 Sentential Logic
 1.1.1 Introduction
 1.1.2 Getting into Sentential Logic
 1.1.3 Compound Sentences (“AND,” “OR,” and “NOT”)
 1.1.4 Compound Sentences
 1.1.5 Equivalence, Tautology, and Contradiction
 1.1.6 De Morgan's Laws
 1.1.7 Tautology
 1.1.8 Logical Sentences from Truth Tables: DNF and CNF
 1.1.9 Disjunctive and Conjunctive Normal Forms
 Problems
 1.2 Conditional and Biconditional Connectives
 1.3 Predicate Logic
 1.4 Mathematical Proofs
 1.5 Proofs in Predicate Logic
 1.6 Proof by Mathematical Induction

1.1 Sentential Logic
 Chapter 2: Sets and Counting
 Chapter 3: Relations

Chapter 4: The Real and Complex Number Systems
 4.1 Construction of the Real Numbers
 4.2 The Complete Ordered Field: The Real Numbers

4.3 Complex Numbers
 4.3.1 An Introductory Tale
 4.3.2 Complex Numbers
 4.3.3 Complex Numbers as an Algebraic Field
 4.3.4 Imaginary Numbers and Two Dimensions
 4.3.5 Polar Coordinates
 4.3.6 Complex Exponential and Euler's Theorem
 4.3.7 Complex Variables in Polar Form
 4.3.8 Basic Arithmetic of Complex Numbers
 4.3.9 Roots and Powers of a Complex Number
 Problems
 Chapter 5: Topology

Chapter 6: Algebra

6.1 Symmetries and Algebraic Systems
 6.1.1 Abstraction and Abstract Algebra
 6.1.2 Symmetries
 6.1.3 Symmetries in Two Dimensions
 6.1.4 Symmetry Transformations
 6.1.5 Symmetries of a Rectangle
 6.1.6 Observations
 6.1.7 Symmetries of an Equilateral Triangle
 6.1.8 Rotation Symmetries of Polyhedra
 6.1.9 Rotation Symmetries of a Cube
 Problems
 6.2 Introduction to the Algebraic Group
 6.3 Permutation Groups
 6.4 Subgroups: Groups Inside a Group
 6.5 Rings and Fields

6.1 Symmetries and Algebraic Systems
 Index
 End User License Agreement
Product information
 Title: Advanced Mathematics
 Author(s):
 Release date: October 2019
 Publisher(s): Wiley
 ISBN: 9781119563518
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