# Mathematical Proofs

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## Introduction to Mathematical Proofs

Author | : Charles Roberts |

Publsiher | : CRC Press |

Total Pages | : 434 |

Release | : 2009-06-24 |

ISBN | : 142006956X |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Mathematical Proofs Book Excerpt:**

Shows How to Read & Write Mathematical ProofsIdeal Foundation for More Advanced Mathematics CoursesIntroduction to Mathematical Proofs: A Transition facilitates a smooth transition from courses designed to develop computational skills and problem solving abilities to courses that emphasize theorem proving. It helps students develop the skills n

## Metamath A Computer Language for Mathematical Proofs

Author | : Norman Megill,David A. Wheeler |

Publsiher | : Lulu.com |

Total Pages | : 250 |

Release | : 2019-06-06 |

ISBN | : 0359702236 |

Category | : Electronic Book |

Language | : EN, FR, DE, ES & NL |

**Metamath A Computer Language for Mathematical Proofs Book Excerpt:**

Metamath is a computer language and an associated computer program for archiving, verifying, and studying mathematical proofs. The Metamath language is simple and robust, with an almost total absence of hard-wired syntax, and we believe that it provides about the simplest possible framework that allows essentially all of mathematics to be expressed with absolute rigor. While simple, it is also powerful; the Metamath Proof Explorer (MPE) database has over 23,000 proven theorems and is one of the top systems in the "Formalizing 100 Theorems" challenge. This book explains the Metamath language and program, with specific emphasis on the fundamentals of the MPE database.

## An Introduction to Mathematical Proofs

Author | : Nicholas A. Loehr |

Publsiher | : CRC Press |

Total Pages | : 396 |

Release | : 2019-11-20 |

ISBN | : 1000709809 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**An Introduction to Mathematical Proofs Book Excerpt:**

An Introduction to Mathematical Proofs presents fundamental material on logic, proof methods, set theory, number theory, relations, functions, cardinality, and the real number system. The text uses a methodical, detailed, and highly structured approach to proof techniques and related topics. No prerequisites are needed beyond high-school algebra. New material is presented in small chunks that are easy for beginners to digest. The author offers a friendly style without sacrificing mathematical rigor. Ideas are developed through motivating examples, precise definitions, carefully stated theorems, clear proofs, and a continual review of preceding topics. Features Study aids including section summaries and over 1100 exercises Careful coverage of individual proof-writing skills Proof annotations and structural outlines clarify tricky steps in proofs Thorough treatment of multiple quantifiers and their role in proofs Unified explanation of recursive definitions and induction proofs, with applications to greatest common divisors and prime factorizations About the Author: Nicholas A. Loehr is an associate professor of mathematics at Virginia Technical University. He has taught at College of William and Mary, United States Naval Academy, and University of Pennsylvania. He has won many teaching awards at three different schools. He has published over 50 journal articles. He also authored three other books for CRC Press, including Combinatorics, Second Edition, and Advanced Linear Algebra.

## LOGIC SETS AND THE TECHNIQUES OF MATHEMATICAL PROOFS

Author | : Brahima MBODJE, Ph.D. |

Publsiher | : Author House |

Total Pages | : 358 |

Release | : 2011-06-30 |

ISBN | : 1463429665 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**LOGIC SETS AND THE TECHNIQUES OF MATHEMATICAL PROOFS Book Excerpt:**

As its title indicates, this book is about logic, sets and mathematical proofs. It is a careful, patient and rigorous introduction for readers with very limited mathematical maturity. It teaches the reader not only how to read a mathematical proof, but also how to write one. To achieve this, we carefully lay out all the various proof methods encountered in mathematical discourse, give their logical justifications, and apply them to the study of topics [such as real numbers, relations, functions, sequences, fine sets, infinite sets, countable sets, uncountable sets and transfinite numbers] whose mastery is important for anyone contemplating advanced studies in mathematics. The book is completely self-contained; since the prerequisites for reading it are only a sound background in high school algebra. Though this book is meant to be a companion specifically for senior high school pupils and college undergraduate students, it will also be of immense value to anyone interested in acquiring the tools and way of thinking of the mathematician.

## Science Of Learning Mathematical Proofs The An Introductory Course

Author | : Elana Reiser |

Publsiher | : World Scientific |

Total Pages | : 244 |

Release | : 2020-11-25 |

ISBN | : 9811225532 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Science Of Learning Mathematical Proofs The An Introductory Course Book Excerpt:**

College students struggle with the switch from thinking of mathematics as a calculation based subject to a problem solving based subject. This book describes how the introduction to proofs course can be taught in a way that gently introduces students to this new way of thinking. This introduction utilizes recent research in neuroscience regarding how the brain learns best. Rather than jumping right into proofs, students are first taught how to change their mindset about learning, how to persevere through difficult problems, how to work successfully in a group, and how to reflect on their learning. With these tools in place, students then learn logic and problem solving as a further foundation.Next various proof techniques such as direct proofs, proof by contraposition, proof by contradiction, and mathematical induction are introduced. These proof techniques are introduced using the context of number theory. The last chapter uses Calculus as a way for students to apply the proof techniques they have learned.

## The History of Mathematical Proof in Ancient Traditions

Author | : Karine Chemla |

Publsiher | : Cambridge University Press |

Total Pages | : 135 |

Release | : 2012-07-05 |

ISBN | : 1139510584 |

Category | : Philosophy |

Language | : EN, FR, DE, ES & NL |

**The History of Mathematical Proof in Ancient Traditions Book Excerpt:**

This radical, profoundly scholarly book explores the purposes and nature of proof in a range of historical settings. It overturns the view that the first mathematical proofs were in Greek geometry and rested on the logical insights of Aristotle by showing how much of that view is an artefact of nineteenth-century historical scholarship. It documents the existence of proofs in ancient mathematical writings about numbers and shows that practitioners of mathematics in Mesopotamian, Chinese and Indian cultures knew how to prove the correctness of algorithms, which are much more prominent outside the limited range of surviving classical Greek texts that historians have taken as the paradigm of ancient mathematics. It opens the way to providing the first comprehensive, textually based history of proof.

## Understanding Mathematical Proof

Author | : John Taylor,Rowan Garnier |

Publsiher | : CRC Press |

Total Pages | : 414 |

Release | : 2016-04-19 |

ISBN | : 1466514914 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Understanding Mathematical Proof Book Excerpt:**

The notion of proof is central to mathematics yet it is one of the most difficult aspects of the subject to teach and master. In particular, undergraduate mathematics students often experience difficulties in understanding and constructing proofs.Understanding Mathematical Proof describes the nature of mathematical proof, explores the various techn

## Proofs in Competition Math Volume 1

Author | : Alexander Toller,Freya Edholm,Dennis Chen |

Publsiher | : Lulu.com |

Total Pages | : 460 |

Release | : 2022 |

ISBN | : 0359714927 |

Category | : Electronic Book |

Language | : EN, FR, DE, ES & NL |

**Proofs in Competition Math Volume 1 Book Excerpt:**

## Introduction to Mathematical Structures and Proofs

Author | : Larry J. Gerstein |

Publsiher | : Springer Science & Business Media |

Total Pages | : 401 |

Release | : 2012-06-05 |

ISBN | : 1461442656 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Mathematical Structures and Proofs Book Excerpt:**

As a student moves from basic calculus courses into upper-division courses in linear and abstract algebra, real and complex analysis, number theory, topology, and so on, a "bridge" course can help ensure a smooth transition. Introduction to Mathematical Structures and Proofs is a textbook intended for such a course, or for self-study. This book introduces an array of fundamental mathematical structures. It also explores the delicate balance of intuition and rigor—and the flexible thinking—required to prove a nontrivial result. In short, this book seeks to enhance the mathematical maturity of the reader. The new material in this second edition includes a section on graph theory, several new sections on number theory (including primitive roots, with an application to card-shuffling), and a brief introduction to the complex numbers (including a section on the arithmetic of the Gaussian integers). Solutions for even numbered exercises are available on springer.com for instructors adopting the text for a course.

## Proof Technology in Mathematics Research and Teaching

Author | : Gila Hanna,David A. Reid,Michael de Villiers |

Publsiher | : Springer Nature |

Total Pages | : 379 |

Release | : 2019-10-02 |

ISBN | : 3030284832 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Proof Technology in Mathematics Research and Teaching Book Excerpt:**

This book presents chapters exploring the most recent developments in the role of technology in proving. The full range of topics related to this theme are explored, including computer proving, digital collaboration among mathematicians, mathematics teaching in schools and universities, and the use of the internet as a site of proof learning. Proving is sometimes thought to be the aspect of mathematical activity most resistant to the influence of technological change. While computational methods are well known to have a huge importance in applied mathematics, there is a perception that mathematicians seeking to derive new mathematical results are unaffected by the digital era. The reality is quite different. Digital technologies have transformed how mathematicians work together, how proof is taught in schools and universities, and even the nature of proof itself. Checking billions of cases in extremely large but finite sets, impossible a few decades ago, has now become a standard method of proof. Distributed proving, by teams of mathematicians working independently on sections of a problem, has become very much easier as digital communication facilitates the sharing and comparison of results. Proof assistants and dynamic proof environments have influenced the verification or refutation of conjectures, and ultimately how and why proof is taught in schools. And techniques from computer science for checking the validity of programs are being used to verify mathematical proofs. Chapters in this book include not only research reports and case studies, but also theoretical essays, reviews of the state of the art in selected areas, and historical studies. The authors are experts in the field.

## Advances in Mathematics Education Research on Proof and Proving

Author | : Andreas J. Stylianides,Guershon Harel |

Publsiher | : Springer |

Total Pages | : 301 |

Release | : 2018-01-10 |

ISBN | : 3319709968 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Advances in Mathematics Education Research on Proof and Proving Book Excerpt:**

This book explores new trends and developments in mathematics education research related to proof and proving, the implications of these trends and developments for theory and practice, and directions for future research. With contributions from researchers working in twelve different countries, the book brings also an international perspective to the discussion and debate of the state of the art in this important area. The book is organized around the following four themes, which reflect the breadth of issues addressed in the book: • Theme 1: Epistemological issues related to proof and proving; • Theme 2: Classroom-based issues related to proof and proving; • Theme 3: Cognitive and curricular issues related to proof and proving; and • Theme 4: Issues related to the use of examples in proof and proving. Under each theme there are four main chapters and a concluding chapter offering a commentary on the theme overall.

## Proof and Proving in Mathematics Education

Author | : Gila Hanna,Michael de Villiers |

Publsiher | : Springer Science & Business Media |

Total Pages | : 468 |

Release | : 2012-06-14 |

ISBN | : 9400721293 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Proof and Proving in Mathematics Education Book Excerpt:**

*THIS BOOK IS AVAILABLE AS OPEN ACCESS BOOK ON SPRINGERLINK* One of the most significant tasks facing mathematics educators is to understand the role of mathematical reasoning and proving in mathematics teaching, so that its presence in instruction can be enhanced. This challenge has been given even greater importance by the assignment to proof of a more prominent place in the mathematics curriculum at all levels. Along with this renewed emphasis, there has been an upsurge in research on the teaching and learning of proof at all grade levels, leading to a re-examination of the role of proof in the curriculum and of its relation to other forms of explanation, illustration and justification. This book, resulting from the 19th ICMI Study, brings together a variety of viewpoints on issues such as: The potential role of reasoning and proof in deepening mathematical understanding in the classroom as it does in mathematical practice. The developmental nature of mathematical reasoning and proof in teaching and learning from the earliest grades. The development of suitable curriculum materials and teacher education programs to support the teaching of proof and proving. The book considers proof and proving as complex but foundational in mathematics. Through the systematic examination of recent research this volume offers new ideas aimed at enhancing the place of proof and proving in our classrooms.

## Proof and the Art of Mathematics

Author | : Joel David Hamkins |

Publsiher | : MIT Press |

Total Pages | : 240 |

Release | : 2020-09-29 |

ISBN | : 0262360934 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Proof and the Art of Mathematics Book Excerpt:**

An introduction to writing proofs, presented through compelling mathematical statements with interesting elementary proofs. This book offers an introduction to the art and craft of proof-writing. The author, a leading research mathematician, presents a series of engaging and compelling mathematical statements with interesting elementary proofs. These proofs capture a wide range of topics, including number theory, combinatorics, graph theory, the theory of games, geometry, infinity, order theory, and real analysis. The goal is to show students and aspiring mathematicians how to write proofs with elegance and precision.

## Introduction to Proof in Abstract Mathematics

Author | : Andrew Wohlgemuth |

Publsiher | : Courier Corporation |

Total Pages | : 385 |

Release | : 2011-02-17 |

ISBN | : 0486478548 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Proof in Abstract Mathematics Book Excerpt:**

Originally published: Philadelphia: Saunders College Pub., c1990.

## Concepts of Proof in Mathematics Philosophy and Computer Science

Author | : Dieter Probst,Peter Schuster |

Publsiher | : Walter de Gruyter GmbH & Co KG |

Total Pages | : 384 |

Release | : 2016-07-25 |

ISBN | : 150150262X |

Category | : Philosophy |

Language | : EN, FR, DE, ES & NL |

**Concepts of Proof in Mathematics Philosophy and Computer Science Book Excerpt:**

A proof is a successful demonstration that a conclusion necessarily follows by logical reasoning from axioms which are considered evident for the given context and agreed upon by the community. It is this concept that sets mathematics apart from other disciplines and distinguishes it as the prototype of a deductive science. Proofs thus are utterly relevant for research, teaching and communication in mathematics and of particular interest for the philosophy of mathematics. In computer science, moreover, proofs have proved to be a rich source for already certified algorithms. This book provides the reader with a collection of articles covering relevant current research topics circled around the concept 'proof'. It tries to give due consideration to the depth and breadth of the subject by discussing its philosophical and methodological aspects, addressing foundational issues induced by Hilbert's Programme and the benefits of the arising formal notions of proof, without neglecting reasoning in natural language proofs and applications in computer science such as program extraction.

## Introduction to Discrete Mathematics via Logic and Proof

Author | : Calvin Jongsma |

Publsiher | : Springer Nature |

Total Pages | : 482 |

Release | : 2019-11-08 |

ISBN | : 3030253589 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Introduction to Discrete Mathematics via Logic and Proof Book Excerpt:**

This textbook introduces discrete mathematics by emphasizing the importance of reading and writing proofs. Because it begins by carefully establishing a familiarity with mathematical logic and proof, this approach suits not only a discrete mathematics course, but can also function as a transition to proof. Its unique, deductive perspective on mathematical logic provides students with the tools to more deeply understand mathematical methodology—an approach that the author has successfully classroom tested for decades. Chapters are helpfully organized so that, as they escalate in complexity, their underlying connections are easily identifiable. Mathematical logic and proofs are first introduced before moving onto more complex topics in discrete mathematics. Some of these topics include: Mathematical and structural induction Set theory Combinatorics Functions, relations, and ordered sets Boolean algebra and Boolean functions Graph theory Introduction to Discrete Mathematics via Logic and Proof will suit intermediate undergraduates majoring in mathematics, computer science, engineering, and related subjects with no formal prerequisites beyond a background in secondary mathematics.

## Conceptions and Consequences of Mathematical Argumentation Justification and Proof

Author | : Kristen N. Bieda,AnnaMarie Conner,Karl W. Kosko,Megan Staples |

Publsiher | : Springer Nature |

Total Pages | : 331 |

Release | : 2022 |

ISBN | : 3030800083 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Conceptions and Consequences of Mathematical Argumentation Justification and Proof Book Excerpt:**

This book aims to advance ongoing debates in the field of mathematics and mathematics education regarding conceptions of argumentation, justification, and proof and the consequences for research and practice when applying particular conceptions of each construct. Through analyses of classroom practice across grade levels using different lenses - particular conceptions of argumentation, justification, and proof - researchers consider the implications of how each conception shapes empirical outcomes. In each section, organized by grade band, authors adopt particular conceptions of argumentation, justification, and proof, and they analyse one data set from each perspective. In addition, each section includes a synthesis chapter from an expert in the field to bring to the fore potential implications, as well as new questions, raised by the analyses. Finally, a culminating section considers the use of each conception across grade bands and data sets.

## Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification

Author | : Klaus Mainzer,Helmut Schwichtenberg,Peter Michael Schuster |

Publsiher | : World Scientific |

Total Pages | : 424 |

Release | : 2021-07-27 |

ISBN | : 9811236496 |

Category | : Mathematics |

Language | : EN, FR, DE, ES & NL |

**Proof And Computation Ii From Proof Theory And Univalent Mathematics To Program Extraction And Verification Book Excerpt:**

This book is for graduate students and researchers, introducing modern foundational research in mathematics, computer science, and philosophy from an interdisciplinary point of view. Its scope includes proof theory, constructive mathematics and type theory, univalent mathematics and point-free approaches to topology, extraction of certified programs from proofs, automated proofs in the automotive industry, as well as the philosophical and historical background of proof theory. By filling the gap between (under-)graduate level textbooks and advanced research papers, the book gives a scholarly account of recent developments and emerging branches of the aforementioned fields.

## Explanation and Proof in Mathematics

Author | : Gila Hanna,Hans Niels Jahnke,Helmut Pulte |

Publsiher | : Springer Science & Business Media |

Total Pages | : 294 |

Release | : 2009-12-04 |

ISBN | : 1441905766 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Explanation and Proof in Mathematics Book Excerpt:**

In the four decades since Imre Lakatos declared mathematics a "quasi-empirical science," increasing attention has been paid to the process of proof and argumentation in the field -- a development paralleled by the rise of computer technology and the mounting interest in the logical underpinnings of mathematics. Explanantion and Proof in Mathematics assembles perspectives from mathematics education and from the philosophy and history of mathematics to strengthen mutual awareness and share recent findings and advances in their interrelated fields. With examples ranging from the geometrists of the 17th century and ancient Chinese algorithms to cognitive psychology and current educational practice, contributors explore the role of refutation in generating proofs, the varied links between experiment and deduction, the use of diagrammatic thinking in addition to pure logic, and the uses of proof in mathematics education (including a critique of "authoritative" versus "authoritarian" teaching styles). A sampling of the coverage: The conjoint origins of proof and theoretical physics in ancient Greece. Proof as bearers of mathematical knowledge. Bridging knowing and proving in mathematical reasoning. The role of mathematics in long-term cognitive development of reasoning. Proof as experiment in the work of Wittgenstein. Relationships between mathematical proof, problem-solving, and explanation. Explanation and Proof in Mathematics is certain to attract a wide range of readers, including mathematicians, mathematics education professionals, researchers, students, and philosophers and historians of mathematics.

## Encyclopedia of Mathematics Education

Author | : Louise Grinstein,Sally I. Lipsey |

Publsiher | : Routledge |

Total Pages | : 912 |

Release | : 2001-03-15 |

ISBN | : 1136787224 |

Category | : Education |

Language | : EN, FR, DE, ES & NL |

**Encyclopedia of Mathematics Education Book Excerpt:**

First published in 2001. Routledge is an imprint of Taylor & Francis, an informa company.