A Transition to Mathematics with Proofs

A Transition to Mathematics with Proofs

by Michael J Cullinane
Released December 2011
Publisher(s): Jones & Bartlett Learning
ISBN: 9781449627799

Read it now on the O’Reilly learning platform with a 10-day free trial.

O’Reilly members get unlimited access to books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.

Start your free trial

Book description

Developed for the "transition" course for mathematics majors moving beyond the primarily procedural methods of their calculus courses toward a more abstract and conceptual environment found in more advanced courses, A Transition to Mathematics with Proofs emphasizes mathematical rigor and helps students learn how to develop and write mathematical proofs. The author takes great care to develop a text that is accessible and readable for students at all levels. It addresses standard topics such as set theory, number system, logic, relations, functions, and induction in at a pace appropriate for a wide range of readers. Throughout early chapters students gradually become aware of the need for rigor, proof, and precision, and mathematical ideas are motivated through examples.

Table of contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Mathematics and Mathematical Activity
    1. 1.1 What Is Mathematics?
    2. 1.2 Mathematical Research and Problem Solving
    3. 1.3 An Example of a Mathematical Research Situation
    4. 1.4 Conjectures and Theorems
    5. 1.5 Methods of Reasoning
    6. 1.6 Why Do We Need Proofs?
    7. 1.7 Mathematical Writing
    8. 1.8 Reading a Mathematics Textbook
    9. Chapter 1 Problems
  7. 2 Sets, Numbers, and Axioms
    1. 2.1 Sets and Numbers from an Intuitive Perspective
    2. 2.2 Set Equality and Set Inclusion
    3. 2.3 Venn Diagrams and Set Operations
    4. 2.4 Undefined Notions and Axioms of Set Theory
    5. 2.5 Axioms for the Real Numbers
    6. Chapter 2 Problems
  8. 3 Elementary Logic
    1. 3.1 Statements and Truth
    2. 3.2 Truth Tables and Statement Forms
    3. 3.3 Logical Equivalence
    4. 3.4 Arguments and Validity
    5. 3.5 Statements Involving Quantifiers
    6. Chapter 3 Problems
  9. 4 Planning and Writing Proofs
    1. 4.1 The Proof-Writing Context
    2. 4.2 Proving an If…Then Statement
    3. 4.3 Proving a For All Statement
    4. 4.4 The Know/Show Approach to Developing Proofs
    5. 4.5 Existence and Uniqueness
    6. 4.6 The Role of Definitions in Creating Proofs
    7. 4.7 Proving and Expressing a Mathematical Equivalence
    8. 4.8 Indirect Methods of Proof
    9. 4.9 Proofs Involving Or
    10. 4.10 A Mathematical Research Situation
    11. Chapter 4 Problems
  10. 5 Relations and Functions
    1. 5.1 Relations
    2. 5.2 Equivalence Relations and Partitions
    3. 5.3 Functions
    4. 5.4 One-to-One Functions, Onto Functions, and Bijections
    5. 5.5 Inverse Relations and Inverse Functions
    6. Chapter 5 Problems
  11. 6 The Natural Numbers, Induction, and Counting
    1. 6.1 Axioms for the Natural Numbers
    2. 6.2 Proof by Induction
    3. 6.3 Recursive Definition and Strong Induction
    4. 6.4 Elementary Number Theory
    5. 6.5 Some Elementary Counting Methods
    6. Chapter 6 Problems
  12. 7 Further Mathematical Explorations
    1. 7.1 Exploring Graph Theory
    2. 7.2 Exploring Groups
    3. 7.3 Exploring Set Cardinality
  13. Index

Product information

  • Title: A Transition to Mathematics with Proofs
  • Author(s): Michael J Cullinane
  • Release date: December 2011
  • Publisher(s): Jones & Bartlett Learning
  • ISBN: 9781449627799