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Theorems, Corollaries, Lemmas, and Methods of Proof

Author : Richard J. Rossi

Publisher : John Wiley & Sons

Page : 338 pages

File Size : 42,14 MB

Release : 2011-10-05

Category : Mathematics

ISBN : 1118030575

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Book Description

A hands-on introduction to the tools needed for rigorous and theoretical mathematical reasoning Successfully addressing the frustration many students experience as they make the transition from computational mathematics to advanced calculus and algebraic structures, Theorems, Corollaries, Lemmas, and Methods of Proof equips students with the tools needed to succeed while providing a firm foundation in the axiomatic structure of modern mathematics. This essential book: Clearly explains the relationship between definitions, conjectures, theorems, corollaries, lemmas, and proofs Reinforces the foundations of calculus and algebra Explores how to use both a direct and indirect proof to prove a theorem Presents the basic properties of real numbers/li> Discusses how to use mathematical induction to prove a theorem Identifies the different types of theorems Explains how to write a clear and understandable proof Covers the basic structure of modern mathematics and the key components of modern mathematics A complete chapter is dedicated to the different methods of proof such as forward direct proofs, proof by contrapositive, proof by contradiction, mathematical induction, and existence proofs. In addition, the author has supplied many clear and detailed algorithms that outline these proofs. Theorems, Corollaries, Lemmas, and Methods of Proof uniquely introduces scratch work as an indispensable part of the proof process, encouraging students to use scratch work and creative thinking as the first steps in their attempt to prove a theorem. Once their scratch work successfully demonstrates the truth of the theorem, the proof can be written in a clear and concise fashion. The basic structure of modern mathematics is discussed, and each of the key components of modern mathematics is defined. Numerous exercises are included in each chapter, covering a wide range of topics with varied levels of difficulty. Intended as a main text for mathematics courses such as Methods of Proof, Transitions to Advanced Mathematics, and Foundations of Mathematics, the book may also be used as a supplementary textbook in junior- and senior-level courses on advanced calculus, real analysis, and modern algebra.



Book of Proof

Author : Richard H. Hammack

Publisher :

Page : 314 pages

File Size : 40,18 MB

Release : 2016-01-01

Category : Mathematics

ISBN : 9780989472111

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Book Description

This book is an introduction to the language and standard proof methods of mathematics. It is a bridge from the computational courses (such as calculus or differential equations) that students typically encounter in their first year of college to a more abstract outlook. It lays a foundation for more theoretical courses such as topology, analysis and abstract algebra. Although it may be more meaningful to the student who has had some calculus, there is really no prerequisite other than a measure of mathematical maturity.



Machine Proofs in Geometry

Author : Shang-Ching Chou

Publisher : World Scientific

Page : 490 pages

File Size : 19,8 MB

Release : 1994

Category : Mathematics

ISBN : 9789810215842

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Book Description

This book reports recent major advances in automated reasoning in geometry. The authors have developed a method and implemented a computer program which, for the first time, produces short and readable proofs for hundreds of geometry theorems.The book begins with chapters introducing the method at an elementary level, which are accessible to high school students; latter chapters concentrate on the main theme: the algorithms and computer implementation of the method.This book brings researchers in artificial intelligence, computer science and mathematics to a new research frontier of automated geometry reasoning. In addition, it can be used as a supplementary geometry textbook for students, teachers and geometers. By presenting a systematic way of proving geometry theorems, it makes the learning and teaching of geometry easier and may change the way of geometry education.



Proofs and Fundamentals

Author : Ethan D. Bloch

Publisher : Springer Science & Business Media

Page : 434 pages

File Size : 30,73 MB

Release : 2013-12-01

Category : Mathematics

ISBN : 1461221307

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Book Description

The aim of this book is to help students write mathematics better. Throughout it are large exercise sets well-integrated with the text and varying appropriately from easy to hard. Basic issues are treated, and attention is given to small issues like not placing a mathematical symbol directly after a punctuation mark. And it provides many examples of what students should think and what they should write and how these two are often not the same.



Proofs and Refutations

Author : Imre Lakatos

Publisher : Cambridge University Press

Page : 190 pages

File Size : 41,20 MB

Release : 1976

Category : Mathematics

ISBN : 9780521290388

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Book Description

Proofs and Refutations is for those interested in the methodology, philosophy and history of mathematics.



Inequalities

Author : Zdravko Cvetkovski

Publisher : Springer Science & Business Media

Page : 439 pages

File Size : 21,96 MB

Release : 2012-01-06

Category : Mathematics

ISBN : 3642237924

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Book Description

This work is about inequalities which play an important role in mathematical Olympiads. It contains 175 solved problems in the form of exercises and, in addition, 310 solved problems. The book also covers the theoretical background of the most important theorems and techniques required for solving inequalities. It is written for all middle and high-school students, as well as for graduate and undergraduate students. School teachers and trainers for mathematical competitions will also gain benefit from this book.



A Concise Handbook of Mathematics, Physics, and Engineering Sciences

Author : Andrei D. Polyanin

Publisher : CRC Press

Page : 1080 pages

File Size : 15,2 MB

Release : 2010-10-18

Category : Mathematics

ISBN : 1439806403

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Book Description

A Concise Handbook of Mathematics, Physics, and Engineering Sciences takes a practical approach to the basic notions, formulas, equations, problems, theorems, methods, and laws that most frequently occur in scientific and engineering applications and university education. The authors pay special attention to issues that many engineers and students



Mathematical Logic

Author : George Tourlakis

Publisher : John Wiley & Sons

Page : 314 pages

File Size : 20,1 MB

Release : 2011-03-01

Category : Mathematics

ISBN : 1118030699

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Book Description

A comprehensive and user-friendly guide to the use of logic in mathematical reasoning Mathematical Logic presents a comprehensive introduction to formal methods of logic and their use as a reliable tool for deductive reasoning. With its user-friendly approach, this book successfully equips readers with the key concepts and methods for formulating valid mathematical arguments that can be used to uncover truths across diverse areas of study such as mathematics, computer science, and philosophy. The book develops the logical tools for writing proofs by guiding readers through both the established "Hilbert" style of proof writing, as well as the "equational" style that is emerging in computer science and engineering applications. Chapters have been organized into the two topical areas of Boolean logic and predicate logic. Techniques situated outside formal logic are applied to illustrate and demonstrate significant facts regarding the power and limitations of logic, such as: Logic can certify truths and only truths. Logic can certify all absolute truths (completeness theorems of Post and Gödel). Logic cannot certify all "conditional" truths, such as those that are specific to the Peano arithmetic. Therefore, logic has some serious limitations, as shown through Gödel's incompleteness theorem. Numerous examples and problem sets are provided throughout the text, further facilitating readers' understanding of the capabilities of logic to discover mathematical truths. In addition, an extensive appendix introduces Tarski semantics and proceeds with detailed proofs of completeness and first incompleteness theorems, while also providing a self-contained introduction to the theory of computability. With its thorough scope of coverage and accessible style, Mathematical Logic is an ideal book for courses in mathematics, computer science, and philosophy at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners who wish to learn how to use logic in their everyday work.



Mathematical Logic through Python

Author : Yannai A. Gonczarowski

Publisher : Cambridge University Press

Page : 286 pages

File Size : 41,10 MB

Release : 2022-07-31

Category : Computers

ISBN : 1108957692

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Book Description

Using a unique pedagogical approach, this text introduces mathematical logic by guiding students in implementing the underlying logical concepts and mathematical proofs via Python programming. This approach, tailored to the unique intuitions and strengths of the ever-growing population of programming-savvy students, brings mathematical logic into the comfort zone of these students and provides clarity that can only be achieved by a deep hands-on understanding and the satisfaction of having created working code. While the approach is unique, the text follows the same set of topics typically covered in a one-semester undergraduate course, including propositional logic and first-order predicate logic, culminating in a proof of Gödel's completeness theorem. A sneak peek to Gödel's incompleteness theorem is also provided. The textbook is accompanied by an extensive collection of programming tasks, code skeletons, and unit tests. Familiarity with proofs and basic proficiency in Python is assumed.



Handbook of Mathematics for Engineers and Scientists

Author : Andrei D. Polyanin

Publisher : CRC Press

Page : 1542 pages

File Size : 19,80 MB

Release : 2006-11-27

Category : Mathematics

ISBN : 1420010514

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Book Description

Covering the main fields of mathematics, this handbook focuses on the methods used for obtaining solutions of various classes of mathematical equations that underlie the mathematical modeling of numerous phenomena and processes in science and technology. The authors describe formulas, methods, equations, and solutions that are frequently used in scientific and engineering applications and present classical as well as newer solution methods for various mathematical equations. The book supplies numerous examples, graphs, figures, and diagrams and contains many results in tabular form, including finite sums and series and exact solutions of differential, integral, and functional equations.