Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Proofs and Ideas: A Prelude to Advanced Mathematics
 
B. Sethuraman California State University, Northridge, CA and Krea University, Sri City, India
Proofs and Ideas
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6514-8
Product Code:  TEXT/68
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-6761-6
Product Code:  TEXT/68.E
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
Softcover ISBN:  978-1-4704-6514-8
eBook: ISBN:  978-1-4704-6761-6
Product Code:  TEXT/68.B
List Price: $170.00 $127.50
MAA Member Price: $127.50 $95.63
AMS Member Price: $127.50 $95.63
Proofs and Ideas
Click above image for expanded view
Proofs and Ideas: A Prelude to Advanced Mathematics
B. Sethuraman California State University, Northridge, CA and Krea University, Sri City, India
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-6514-8
Product Code:  TEXT/68
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
eBook ISBN:  978-1-4704-6761-6
Product Code:  TEXT/68.E
List Price: $85.00
MAA Member Price: $63.75
AMS Member Price: $63.75
Softcover ISBN:  978-1-4704-6514-8
eBook ISBN:  978-1-4704-6761-6
Product Code:  TEXT/68.B
List Price: $170.00 $127.50
MAA Member Price: $127.50 $95.63
AMS Member Price: $127.50 $95.63
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 682021; 334 pp
    MSC: Primary 00;

    Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.

    The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and non-routine.

    This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.

    Readership

    Undergraduate students interested in an introduction to proofs.

  • Table of Contents
     
     
    • Title page
    • Copyright
    • Contents
    • Preface
    • Chapter 1. Introduction
    • 1.1. Further Exercises
    • Chapter 2. The Pigeonhole Principle
    • 2.1. Pigeonhole Principle (PHP)
    • 2.2. PHP Generalized Form
    • 2.3. Further Exercises
    • Chapter 3. Statements
    • 3.1. Statements
    • 3.2. Negation of a Statement
    • 3.3. Compound Statements
    • 3.4. Statements Related to the Conditional
    • 3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations
    • 3.6. Further Exercises
    • Chapter 4. Counting, Combinations
    • 4.1. Fundamental Counting Principle
    • 4.2. Permutations and Combinations
    • 4.3. Binomial Relations and Binomial Theorem
    • 4.4. Further Exercises
    • Chapter 5. Sets and Functions
    • 5.1. Sets
    • 5.2. Equality of Sets, Subsets, Supersets
    • 5.3. New Sets From Old
    • 5.4. Functions Between Sets
    • 5.5. Composition of Functions, Inverses
    • 5.6. Examples of Some Sets Commonly Occurring in Mathematics
    • 5.7. Further Exercises
    • Chapter 6. Interlude: So, How to Prove It? An Essay
    • Chapter 7. Induction
    • 7.1. Principle of Induction
    • 7.2. Another Form of the Induction Principle
    • 7.3. Further Exercises
    • 7.4. Notes
    • Chapter 8. Cardinality of Sets
    • 8.1. Finite and Infinite Sets, Countability, Uncountability
    • 8.2. Cardinalities of Q and R
    • 8.3. The Schröder-Bernstein Theorem
    • 8.4. Cantor Set
    • 8.5. Further Exericses
    • Chapter 9. Equivalence Relations
    • 9.1. Relations, Equivalence Relations, Equivalence Classes
    • 9.2. Examples
    • 9.3. Further Exercises
    • Chapter 10. Unique Prime Factorization in the Integers
    • 10.1. Notion of Divisibility
    • 10.2. Greatest Common Divisor, Relative Primeness
    • 10.3. Proof of Unique Prime Factorization Theorem
    • 10.4. Some Consequences of the Unique Prime Factorization Theorem
    • 10.5. Further Exercises
    • Chapter 11. Sequences, Series, Continuity, Limits
    • 11.1. Sequences
    • 11.2. Convergence
    • 11.3. Continuity of Functions
    • 11.4. Limits of Functions
    • 11.5. Relation between limits and continuity
    • 11.6. Series
    • 11.7. Further Exercises
    • Chapter 12. The Completeness of R
    • 12.1. Least Upper Bound Property (LUB)
    • 12.2. Greatest Lower Bound Property
    • 12.3. Archimedean Property
    • 12.4. Monotone Convergence Theorem
    • 12.5. Bolzano-Weierstrass Theorem
    • 12.6. Nested Intervals Theorem
    • 12.7. Cauchy sequences
    • 12.8. Convergence of Series
    • 12.9. 𝑛-th roots of positive real numbers
    • 12.10. Further Exercises
    • Notes
    • Chapter 13. Groups and Symmetry
    • 13.1. Symmetries of an equilateral triangle
    • 13.2. Symmetries of a square
    • 13.3. Symmetries of an 𝑛-element set
    • Groups
    • 13.4. Subgroups
    • 13.5. Cosets, Lagrange’s Theorem
    • 13.6. Symmetry
    • 13.7. Isomorphisms Between Groups
    • 13.8. Further Exercises
    • Chapter 14. Graphs: An Introduction
    • 14.1. Königsberg Bridge Problem and Graphs
    • 14.2. Walks, Paths, Trails, Connectedness
    • 14.3. Existence of Eulerian Trails and Circuits: Sufficiency
    • 14.4. Further Exercises
    • Index
  • Reviews
     
     
    • It is well suited for self-study, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.

      Adhemar François Bultheel, zbMATH
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 682021; 334 pp
MSC: Primary 00;

Proofs and Ideas serves as a gentle introduction to advanced mathematics for students who previously have not had extensive exposure to proofs. It is intended to ease the student's transition from algorithmic mathematics to the world of mathematics that is built around proofs and concepts.

The spirit of the book is that the basic tools of abstract mathematics are best developed in context and that creativity and imagination are at the core of mathematics. So, while the book has chapters on statements and sets and functions and induction, the bulk of the book focuses on core mathematical ideas and on developing intuition. Along with chapters on elementary combinatorics and beginning number theory, this book contains introductory chapters on real analysis, group theory, and graph theory that serve as gentle first exposures to their respective areas. The book contains hundreds of exercises, both routine and non-routine.

This book has been used for a transition to advanced mathematics courses at California State University, Northridge, as well as for a general education course on mathematical reasoning at Krea University, India.

Readership

Undergraduate students interested in an introduction to proofs.

  • Title page
  • Copyright
  • Contents
  • Preface
  • Chapter 1. Introduction
  • 1.1. Further Exercises
  • Chapter 2. The Pigeonhole Principle
  • 2.1. Pigeonhole Principle (PHP)
  • 2.2. PHP Generalized Form
  • 2.3. Further Exercises
  • Chapter 3. Statements
  • 3.1. Statements
  • 3.2. Negation of a Statement
  • 3.3. Compound Statements
  • 3.4. Statements Related to the Conditional
  • 3.5. Remarks on the Implies Statement: Alternative Phrasing, Negations
  • 3.6. Further Exercises
  • Chapter 4. Counting, Combinations
  • 4.1. Fundamental Counting Principle
  • 4.2. Permutations and Combinations
  • 4.3. Binomial Relations and Binomial Theorem
  • 4.4. Further Exercises
  • Chapter 5. Sets and Functions
  • 5.1. Sets
  • 5.2. Equality of Sets, Subsets, Supersets
  • 5.3. New Sets From Old
  • 5.4. Functions Between Sets
  • 5.5. Composition of Functions, Inverses
  • 5.6. Examples of Some Sets Commonly Occurring in Mathematics
  • 5.7. Further Exercises
  • Chapter 6. Interlude: So, How to Prove It? An Essay
  • Chapter 7. Induction
  • 7.1. Principle of Induction
  • 7.2. Another Form of the Induction Principle
  • 7.3. Further Exercises
  • 7.4. Notes
  • Chapter 8. Cardinality of Sets
  • 8.1. Finite and Infinite Sets, Countability, Uncountability
  • 8.2. Cardinalities of Q and R
  • 8.3. The Schröder-Bernstein Theorem
  • 8.4. Cantor Set
  • 8.5. Further Exericses
  • Chapter 9. Equivalence Relations
  • 9.1. Relations, Equivalence Relations, Equivalence Classes
  • 9.2. Examples
  • 9.3. Further Exercises
  • Chapter 10. Unique Prime Factorization in the Integers
  • 10.1. Notion of Divisibility
  • 10.2. Greatest Common Divisor, Relative Primeness
  • 10.3. Proof of Unique Prime Factorization Theorem
  • 10.4. Some Consequences of the Unique Prime Factorization Theorem
  • 10.5. Further Exercises
  • Chapter 11. Sequences, Series, Continuity, Limits
  • 11.1. Sequences
  • 11.2. Convergence
  • 11.3. Continuity of Functions
  • 11.4. Limits of Functions
  • 11.5. Relation between limits and continuity
  • 11.6. Series
  • 11.7. Further Exercises
  • Chapter 12. The Completeness of R
  • 12.1. Least Upper Bound Property (LUB)
  • 12.2. Greatest Lower Bound Property
  • 12.3. Archimedean Property
  • 12.4. Monotone Convergence Theorem
  • 12.5. Bolzano-Weierstrass Theorem
  • 12.6. Nested Intervals Theorem
  • 12.7. Cauchy sequences
  • 12.8. Convergence of Series
  • 12.9. 𝑛-th roots of positive real numbers
  • 12.10. Further Exercises
  • Notes
  • Chapter 13. Groups and Symmetry
  • 13.1. Symmetries of an equilateral triangle
  • 13.2. Symmetries of a square
  • 13.3. Symmetries of an 𝑛-element set
  • Groups
  • 13.4. Subgroups
  • 13.5. Cosets, Lagrange’s Theorem
  • 13.6. Symmetry
  • 13.7. Isomorphisms Between Groups
  • 13.8. Further Exercises
  • Chapter 14. Graphs: An Introduction
  • 14.1. Königsberg Bridge Problem and Graphs
  • 14.2. Walks, Paths, Trails, Connectedness
  • 14.3. Existence of Eulerian Trails and Circuits: Sufficiency
  • 14.4. Further Exercises
  • Index
  • It is well suited for self-study, but several possible selections of the chapters are suggested to set up courses of varying lengths. No special prerequisites are needed besides a basic mathematical formation and a certain desire to move to a higher mathematical level.

    Adhemar François Bultheel, zbMATH
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.