The Nuts and Bolts of Proofs
An Introduction to Mathematical Proofs
- 5th Edition - January 5, 2023
- Imprint: Academic Press
- Author: Antonella Cupillari
- Language: English
- Paperback ISBN:9 7 8 - 0 - 3 2 3 - 9 9 0 2 0 - 2
- eBook ISBN:9 7 8 - 0 - 3 2 3 - 9 9 0 2 1 - 9
The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and how they work. The book offers technique… Read more
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Request a sales quoteThe Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs, Fifth Edition provides basic logic of mathematical proofs and how they work. The book offers techniques for both reading and writing proofs, discusses techniques in proving if/then statements by contrapositive and proofing by contradiction, includes the negation statement, and/or, examines various theorems, such as the if and only-if, equivalence theorems, existence theorems, and the uniqueness theorems. In addition, the use of counter examples, mathematical induction, composite statements including multiple hypothesis and multiple conclusions, and equality of numbers are also covered. The book also provides mathematical topics for practicing proof techniques.
Included here are the Cartesian products, indexed families, functions, and relations. The last chapter of the book provides review exercises on various topics. Undergraduate students in engineering and physical science will find this book accessible as well as invaluable.
- Jumps right in with the needed vocabulary to get students thinking like mathematicians from the beginning
- Offers a large variety of examples and problems with solutions for students to work through on their own
- Includes a collection of exercises without solutions to help instructors prepare assignments
- Contains an extensive list of basic mathematical definitions and concepts needed in abstract mathematics
Upper-level undergraduates in higher/discrete/advanced mathematics, engineering, and physical sciences
- Cover image
- Title page
- Table of Contents
- Copyright
- Dedication
- Preface
- List of Symbols
- Chapter 1. Getting Started
- Introduction and Basic Terminology
- General Suggestions
- Chapter 2. Basic Techniques to Prove If/Then Statements
- What Does “If/Then” Mean?
- Direct Proof
- The Negation of a Statement; AND/OR
- Proof by Contrapositive
- Proof by Contradiction
- Chapter 3. Special Kinds of Theorems
- “If and Only If” or Equivalence Theorems
- Use of Counterexamples
- Mathematical Induction
- Existence Theorems
- Uniqueness Theorems
- Composite Statements
- Equality of Numbers
- Chapter 4. Some Mathematical Topics on Which to Practice Proof Techniques
- Basic Set Theory and Indexed Families
- Power Sets or Parts of a set
- Cartesian Product of Sets
- Indexed Families of Sets
- About Functions
- Composition of Functions
- A Little More About Functions and Sets
- Relations
- Most Common Properties of Relations
- More About Equivalence Relations
- A Special Relation and More Facts About Equivalence Classes
- The Basics of Groups
- Some Properties of Binary Operations
- Special Elements
- When the Properties Fit Together
- Group Sizes and Structures
- Groups (mod m) and Arithmetic (mod m)
- Permutations and Symmetric Groups
- Isomorphism and Subgroups
- An Important Theorem
- Limits
- Getting Closer
- Functions and Limits
- Cardinality and Sizes of Infinity
- Chapter 5. Review Exercises
- Exercises Without Solutions
- Collection of “Proofs”
- Solutions for the Exercises at the End of the Sections and the Review Exercises
- Chapter 2 Basic Techniques to Prove If/Then Statements
- Chapter 3 Special Kinds of Theorems
- Chapter 4 Some Mathematical Topics on Which to Practice Proof Techniques
- Solutions for the Review Exercises
- Index
- Edition: 5
- Published: January 5, 2023
- No. of pages (Paperback): 416
- No. of pages (eBook): 416
- Imprint: Academic Press
- Language: English
- Paperback ISBN: 9780323990202
- eBook ISBN: 9780323990219
AC
Antonella Cupillari
Antonella Cupillari is an associate professor of mathematics at Pennsylvania State Erie in Behrend College. She received her Laurea in Mathematics in Italy, and her M.A. and Ph.D. at the State University of New York at Albany. She has been a participant in the Mathematical Association of America/National Science Foundation Institute on the "History of Mathematics and Its Use in Teaching." Cupillari is the author of several papers in analysis, mathematics education, and the history of mathematics. She is also the author of the first edition of The Nuts and Bolts of Proofs.
Affiliations and expertise
Associate Professor of Mathematics, Pennsylvania State Erie in Behrend College, Erie, PA, USARead The Nuts and Bolts of Proofs on ScienceDirect