Advanced Mathematics

Advanced Mathematics

by Stanley J. Farlow
Released October 2019
Publisher(s): Wiley
ISBN: 9781119563518

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Book description

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 By avoiding outdated methods and countless pages of theorems and proofs, this innovative textbook prompts students to think about the ideas presented in an enjoyable, constructive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

Table of contents

  1. Cover
  2. Preface
  3. Possible Beneficial Audiences
  4. Wow Factors of the Book
  5. Chapter by Chapter (the nitty‐gritty)
  6. Note to the Reader
    1. Keeping a Scholarly Journal
  7. About the Companion Website
  8. Chapter 1: Logic and Proofs
    1. 1.1 Sentential Logic
      1. 1.1.1 Introduction
      2. 1.1.2 Getting into Sentential Logic
      3. 1.1.3 Compound Sentences (“AND,” “OR,” and “NOT”)
      4. 1.1.4 Compound Sentences
      5. 1.1.5 Equivalence, Tautology, and Contradiction
      6. 1.1.6 De Morgan's Laws
      7. 1.1.7 Tautology
      8. 1.1.8 Logical Sentences from Truth Tables: DNF and CNF
      9. 1.1.9 Disjunctive and Conjunctive Normal Forms
      10. Problems
    2. 1.2 Conditional and Biconditional Connectives
      1. 1.2.1 The Conditional Sentence
      2. 1.2.2 Understanding the Conditional Sentence
      3. 1.2.3 Converse, Inverse, and the Contrapositive
      4. 1.2.4 Law of the Syllogism
      5. 1.2.5 A Useful Equivalence for the Implication
      6. 1.2.6 The Biconditional
      7. Problems
    3. 1.3 Predicate Logic
      1. 1.3.1 Introduction
      2. 1.3.2 Existential and Universal Quantifiers
      3. 1.3.3 More than One Variable in a Proposition
      4. 1.3.4 Order Matters
      5. 1.3.5 Negation of Quantified Propositions
      6. 1.3.6 Conjunctions and Disjunctions in Predicate Logic
      7. Problems
    4. 1.4 Mathematical Proofs
      1. 1.4.1 Introduction
      2. 1.4.2 Types of Proofs
      3. 1.4.3 Analysis of Proof Techniques
      4. 1.4.4 Modus Operandi for Proving Theorems
      5. 1.4.5 Necessary and Sufficient Conditions (NASC)
      6. Problems
    5. 1.5 Proofs in Predicate Logic
      1. 1.5.1 Introduction
      2. 1.5.2 Proofs Involving Quantifiers
      3. 1.5.3 Proofs by Contradiction for Quantifiers
      4. 1.5.4 Unending Interesting Properties of Numbers
      5. 1.5.5 Unique Existential Quantification ∃!
      6. Problems
    6. 1.6 Proof by Mathematical Induction
      1. 1.6.1 Introduction
      2. 1.6.2 Mathematical Induction
      3. 1.6.3 Strong Induction
      4. Problems
  9. Chapter 2: Sets and Counting
    1. 2.1 Basic Operations of Sets
      1. 2.1.1 Sets and Membership
      2. 2.1.2 Universe, Subset, Equality, Complement, Empty Set
      3. 2.1.3 Union, Intersection, and Difference of Sets
      4. 2.1.4 Venn Diagrams of Various Sets
      5. 2.1.5 Relation Between Sets and Logic
      6. 2.1.6 De Morgan's Laws for Sets
      7. 2.1.7 Sets, Logic, and Arithmetic
      8. Problems
    2. 2.2 Families of Sets
      1. 2.2.1 Introduction
      2. 2.2.2 Extended Laws for Sets
      3. 2.2.3 Topologies on a Set
      4. Problems
    3. 2.3 Counting: The Art of Enumeration
      1. 2.3.1 Introduction
      2. 2.3.2 Multiplication Principle
      3. 2.3.3 Permutations
      4. 2.3.4 Permutations of Racers
      5. 2.3.5 Distinguishable Permutations
      6. 2.3.6 Combinations
      7. 2.3.7 The Pigeonhole Principle
      8. Problems
    4. 2.4 Cardinality of Sets
      1. 2.4.1 Introduction
      2. 2.4.2 Cardinality, Equivalence, Finite, and Infinite
      3. 2.4.3 Major Result Comparing Sizes of Finite Sets
      4. 2.4.4 Countably Infinite Sets
      5. Problems
    5. 2.5 Uncountable Sets
      1. 2.5.1 Introduction
      2. 2.5.2 Cantor's Surprise
      3. Problems
    6. 2.6 Larger Infinities and the ZFC Axioms
      1. 2.6.1 Cantor's Discovery of Larger Sets
      2. 2.6.2 The Cantor–Bernstein Theorem
      3. 2.6.3 The Continuum Hypothesis
      4. 2.6.4 Need for Axioms in Set Theory
      5. 2.6.5 The Zermelo–Fraenkel Axioms
      6. 2.6.6 Comments on the AC
      7. 2.6.7 Axiom of Choice ⇔ Well‐Ordering Principle
      8. Problems
  10. Chapter 3: Relations
    1. 3.1 Relations
      1. 3.1.1 Introduction and the Cartesian Product
      2. 3.1.2 Relations
      3. 3.1.3 Visualization of Relations with Directed Graphs
      4. 3.1.4 Domain and Range of a Relation
      5. 3.1.5 Inverses and Compositions
      6. 3.1.6 Composition of Relations
      7. Problems
    2. 3.2 Order Relations
      1. 3.2.1 Let There Be Order
      2. 3.2.2 Total Order and Symmetric Relations
      3. 3.2.3 Symmetric Relation
      4. 3.2.4 Hasse Diagrams and Directed Graphs
      5. 3.2.5 Upper Bounds, Lower Bounds, glb, and lub
      6. Problems
    3. 3.3 Equivalence Relations
      1. 3.3.1 Introduction
      2. 3.3.2 Partition of a Set
      3. 3.3.3 The Partitioning Property of the Equivalence Relation
      4. 3.3.4 Counting Partitions
      5. 3.3.5 Modular Arithmetic
      6. Problems
    4. 3.4 The Function Relation
      1. 3.4.1 Introduction
      2. 3.4.2 Relation Definition of a Function
      3. 3.4.3 Composition of Functions
      4. 3.4.4 Inverse Functions
      5. Problems
    5. 3.5 Image of a Set
      1. 3.5.1 Introduction
      2. 3.5.2 Images of Intersections and Unions
      3. Problems
  11. Chapter 4: The Real and Complex Number Systems
    1. 4.1 Construction of the Real Numbers
      1. 4.1.1 Introduction
      2. 4.1.2 The Building of the Real Numbers
      3. 4.1.3 Construction of the Integers: ℕ → ℤ
      4. 4.1.4 Construction of the Rationals: ℤ → ℚ
      5. 4.1.5 How to Define Real Numbers
      6. 4.1.6 How Dedekind Cuts Define the Real Numbers
      7. 4.1.7 Arithmetic of the Real Numbers
      8. Problems
    2. 4.2 The Complete Ordered Field: The Real Numbers
      1. 4.2.1 Introduction
      2. 4.2.2 Arithmetic Axioms for Real Numbers
      3. 4.2.3 Conventions and Notation
      4. 4.2.4 Fields Other than ℝ
      5. 4.2.5 Ordered Fields
      6. 4.2.6 The Completeness Axiom
      7. 4.2.7 Least Upper Bound and Greatest Lower Bounds
      8. Problems
    3. 4.3 Complex Numbers
      1. 4.3.1 An Introductory Tale
      2. 4.3.2 Complex Numbers
      3. 4.3.3 Complex Numbers as an Algebraic Field
      4. 4.3.4 Imaginary Numbers and Two Dimensions
      5. 4.3.5 Polar Coordinates
      6. 4.3.6 Complex Exponential and Euler's Theorem
      7. 4.3.7 Complex Variables in Polar Form
      8. 4.3.8 Basic Arithmetic of Complex Numbers
      9. 4.3.9 Roots and Powers of a Complex Number
      10. Problems
  12. Chapter 5: Topology
    1. 5.1 Introduction to Graph Theory
      1. 5.1.1 Introduction
      2. 5.1.2 Glossary of Important Concepts in Graph Theory
      3. 5.1.3 Euler Paths and Circuits
      4. 5.1.4 Return to Konigsberg
      5. 5.1.5 Weighted Graphs
      6. 5.1.6 Euler's Characteristic for Planar Graphs
      7. Problems
    2. 5.2 Directed Graphs
      1. 5.2.1 Introduction
      2. 5.2.2 Tournament Graphs (Dominance Graphs)
      3. 5.2.3 Dominance Graphs in Social Networking
      4. 5.2.4 PageRank System
      5. 5.2.5 Dynamic Programming
      6. Problems
    3. 5.3 Geometric Topology
      1. 5.3.1 Introduction
      2. 5.3.2 Topological Equivalent Objects
      3. 5.3.3 Homeomorphisms as Equivalence Relations
      4. 5.3.4 Topological Invariants
      5. 5.3.5 Euler Characteristic for Graphs, Polyhedra, and Surfaces
      6. 5.3.6 The Euler Characteristic of a Surface
      7. Problems
    4. 5.4 Point‐Set Topology on the Real Line
      1. 5.4.1 Introduction
      2. 5.4.2 Interior, Exterior, and Boundary of a Set
      3. 5.4.3 Interiors, Boundaries, and Exteriors of Common Sets
      4. 5.4.4 Limit Points
      5. 5.4.5 Closed Sets Contain Their Limit Points
      6. 5.4.6 Topological Spaces
      7. 5.4.7 Calculus Without Topology Is No Calculus
      8. Problems
  13. Chapter 6: Algebra
    1. 6.1 Symmetries and Algebraic Systems
      1. 6.1.1 Abstraction and Abstract Algebra
      2. 6.1.2 Symmetries
      3. 6.1.3 Symmetries in Two Dimensions
      4. 6.1.4 Symmetry Transformations
      5. 6.1.5 Symmetries of a Rectangle
      6. 6.1.6 Observations
      7. 6.1.7 Symmetries of an Equilateral Triangle
      8. 6.1.8 Rotation Symmetries of Polyhedra
      9. 6.1.9 Rotation Symmetries of a Cube
      10. Problems
    2. 6.2 Introduction to the Algebraic Group
      1. 6.2.1 Basics of a Group
      2. 6.2.2 Binary Operations and the Group
      3. 6.2.3 Cayley Table
      4. 6.2.4 Cyclic Groups: Modular Arithmetic
      5. 6.2.5 Isomorphic Groups: Groups that are the Same
      6. 6.2.6 Dihedral Groups: Symmetries of Regular Polygons
      7. 6.2.7 Multiplying Groups
      8. Problems
    3. 6.3 Permutation Groups
      1. 6.3.1 Permutations and Their Products
      2. 6.3.2 Inverses of Permutations
      3. 6.3.3 Cycle Notation for Permutations
      4. 6.3.4 Products of Permutations in Cycle Notation
      5. 6.3.5 Transpositions
      6. 6.3.6 Symmetric Group Sn
      7. 6.3.7 Symmetric Group S3
      8. 6.3.8 Alternating Group
      9. Problems
    4. 6.4 Subgroups: Groups Inside a Group
      1. 6.4.1 Introduction
      2. 6.4.2 Subgroups of the Klein Four‐Group
      3. 6.4.3 Test of Subgroups
      4. 6.4.4 Subgroups of Cyclic Groups
      5. 6.4.5 Cosets and the Quotient Group
      6. Problems
    5. 6.5 Rings and Fields
      1. 6.5.1 Introduction to Rings
      2. 6.5.2 Common Rings
      3. 6.5.3 Algebraic Fields
      4. 6.5.4 Finite Fields
      5. Problems
  14. Index
  15. End User License Agreement

Product information

  • Title: Advanced Mathematics
  • Author(s): Stanley J. Farlow
  • Release date: October 2019
  • Publisher(s): Wiley
  • ISBN: 9781119563518