Advanced Mathematics

Name: Advanced Mathematics | A Transitional Reference
Brand: Wiley
Price: 101.99 EUR
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A Transitional Reference

Stanley J. Farlow(Autor*in)

Wiley (Verlag)

1. Auflage

Erschienen am 10. September 2019

480 Seiten

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978-1-119-56348-8 (ISBN)

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Beschreibung

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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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

Weitere Details

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Inhalt

Intro
Title Page
Copyright Page
Contents
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
1.2.1 The Conditional Sentence
1.2.2 Understanding the Conditional Sentence
1.2.3 Converse, Inverse, and the Contrapositive
1.2.4 Law of the Syllogism
1.2.5 A Useful Equivalence for the Implication
1.2.6 The Biconditional
Problems
1.3 Predicate Logic
1.3.1 Introduction
1.3.2 Existential and Universal Quantifiers
1.3.3 More than One Variable in a Proposition
1.3.4 Order Matters
1.3.5 Negation of Quantified Propositions
1.3.6 Conjunctions and Disjunctions in Predicate Logic
Problems
1.4 Mathematical Proofs
1.4.1 Introduction
1.4.2 Types of Proofs
1.4.3 Analysis of Proof Techniques
1.4.4 Modus Operandi for Proving Theorems
1.4.5 Necessary and Sufficient Conditions (NASC)
Problems
1.5 Proofs in Predicate Logic
1.5.1 Introduction
1.5.2 Proofs Involving Quantifiers
1.5.3 Proofs by Contradiction for Quantifiers
1.5.4 Unending Interesting Properties of Numbers
1.5.5 Unique Existential Quantification !
Problems
1.6 Proof by Mathematical Induction
1.6.1 Introduction
1.6.2 Mathematical Induction
1.6.3 Strong Induction
Problems
Chapter 2 Sets and Counting
2.1 Basic Operations of Sets
2.1.1 Sets and Membership
2.1.2 Universe, Subset, Equality, Complement, Empty Set
2.1.3 Union, Intersection, and Difference of Sets
2.1.4 Venn Diagrams of Various Sets
2.1.5 Relation Between Sets and Logic
2.1.6 De Morgan's Laws for Sets
2.1.7 Sets, Logic, and Arithmetic
Problems
2.2 Families of Sets
2.2.1 Introduction
2.2.2 Extended Laws for Sets
2.2.3 Topologies on a Set
Problems
2.3 Counting: The Art of Enumeration
2.3.1 Introduction
2.3.2 Multiplication Principle
2.3.3 Permutations
2.3.4 Permutations of Racers
2.3.5 Distinguishable Permutations
2.3.6 Combinations
2.3.7 The Pigeonhole Principle
Problems
2.4 Cardinality of Sets
2.4.1 Introduction
2.4.2 Cardinality, Equivalence, Finite, and Infinite
2.4.3 Major Result Comparing Sizes of Finite Sets
2.4.4 Countably Infinite Sets
Problems
2.5 Uncountable Sets
2.5.1 Introduction
2.5.2 Cantor's Surprise
Problems
2.6 Larger Infinities and the ZFC Axioms
2.6.1 Cantor's Discovery of Larger Sets
2.6.2 The Cantor-Bernstein Theorem
2.6.3 The Continuum Hypothesis
2.6.4 Need for Axioms in Set Theory
2.6.5 The Zermelo-Fraenkel Axioms
2.6.6 Comments on the AC
2.6.7 Axiom of Choice Well-Ordering Principle
Problems
Chapter 3 Relations
3.1 Relations
3.1.1 Introduction and the Cartesian Product
3.1.2 Relations
3.1.3 Visualization of Relations with Directed Graphs
3.1.4 Domain and Range of a Relation
3.1.5 Inverses and Compositions
3.1.6 Composition of Relations
Problems
3.2 Order Relations
3.2.1 Let There Be Order
3.2.2 Total Order and Symmetric Relations
3.2.3 Symmetric Relation
3.2.4 Hasse Diagrams and Directed Graphs
3.2.5 Upper Bounds, Lower Bounds, glb, and lub
Problems
3.3 Equivalence Relations
3.3.1 Introduction
3.3.2 Partition of a Set
3.3.3 The Partitioning Property of the Equivalence Relation
3.3.4 Counting Partitions
3.3.5 Modular Arithmetic
Problems
3.4 The Function Relation
3.4.1 Introduction
3.4.2 Relation Definition of a Function
3.4.3 Composition of Functions
3.4.4 Inverse Functions
Problems
3.5 Image of a Set
3.5.1 Introduction
3.5.2 Images of Intersections and Unions
Problems
Chapter 4 The Real and Complex Number Systems
4.1 Construction of the Real Numbers
4.1.1 Introduction
4.1.2 The Building of the Real Numbers
4.1.3 Construction of the Integers:
4.1.4 Construction of the Rationals: Z Q
4.1.5 How to Define Real Numbers
4.1.6 How Dedekind Cuts Define the Real Numbers
4.1.7 Arithmetic of the Real Numbers
Problems
4.2 The Complete Ordered Field: The Real Numbers
4.2.1 Introduction
4.2.2 Arithmetic Axioms for Real Numbers
4.2.3 Conventions and Notation
4.2.4 Fields Other than
4.2.5 Ordered Fields
4.2.6 The Completeness Axiom
4.2.7 Least Upper Bound and Greatest Lower Bounds
Problems
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
5.1 Introduction to Graph Theory
5.1.1 Introduction
5.1.2 Glossary of Important Concepts in Graph Theory
5.1.3 Euler Paths and Circuits
5.1.4 Return to Konigsberg
5.1.5 Weighted Graphs
5.1.6 Euler's Characteristic for Planar Graphs
Problems
5.2 Directed Graphs
5.2.1 Introduction
5.2.2 Tournament Graphs (Dominance Graphs)
5.2.3 Dominance Graphs in Social Networking
5.2.4 PageRank System
5.2.5 Dynamic Programming
Problems
5.3 Geometric Topology
5.3.1 Introduction
5.3.2 Topological Equivalent Objects
5.3.3 Homeomorphisms as Equivalence Relations
5.3.4 Topological Invariants
5.3.5 Euler Characteristic for Graphs, Polyhedra, and Surfaces
5.3.6 The Euler Characteristic of a Surface
Problems
5.4 Point-Set Topology on the Real Line
5.4.1 Introduction
5.4.2 Interior, Exterior, and Boundary of a Set
5.4.3 Interiors, Boundaries, and Exteriors of Common Sets
5.4.4 Limit Points
5.4.5 Closed Sets Contain Their Limit Points
5.4.6 Topological Spaces
5.4.7 Calculus Without Topology Is No Calculus
Problems
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
6.2.1 Basics of a Group
6.2.2 Binary Operations and the Group
6.2.3 Cayley Table
6.2.4 Cyclic Groups: Modular Arithmetic
6.2.5 Isomorphic Groups: Groups that are the Same
6.2.6 Dihedral Groups: Symmetries of Regular Polygons
6.2.7 Multiplying Groups
Problems
6.3 Permutation Groups
6.3.1 Permutations and Their Products
6.3.2 Inverses of Permutations
6.3.3 Cycle Notation for Permutations
6.3.4 Products of Permutations in Cycle Notation
6.3.5 Transpositions
6.3.6 Symmetric Group Sn
6.3.7 Symmetric Group S3
6.3.8 Alternating Group
Problems
6.4 Subgroups: Groups Inside a Group
6.4.1 Introduction
6.4.2 Subgroups of the Klein Four-Group
6.4.3 Test of Subgroups
6.4.4 Subgroups of Cyclic Groups
6.4.5 Cosets and the Quotient Group
Problems
6.5 Rings and Fields
6.5.1 Introduction to Rings
6.5.2 Common Rings
6.5.3 Algebraic Fields
6.5.4 Finite Fields
Problems
Index
EULA

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