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Mathematical Proofs

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Logic Sets and the Techniques of Mathematical Proofs

Logic  Sets and the Techniques of Mathematical Proofs Book
Author : Brahima Mbodje Ph. D.
Publisher : AuthorHouse
Release : 2011-06-01
ISBN : 1463429673
Language : En, Es, Fr & De

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Book Description :

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.

Metamath A Computer Language for Mathematical Proofs

Metamath  A Computer Language for Mathematical Proofs Book
Author : Norman Megill,David A. Wheeler
Publisher : Lulu.com
Release : 2019-06-06
ISBN : 0359702236
Language : En, Es, Fr & De

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Book Description :

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.

Mathematical Proofs

Mathematical Proofs Book
Author : Gary Chartrand,Albert D. Polimeni,Ping Zhang
Publisher : Addison-Wesley Longman
Release : 2008
ISBN :
Language : En, Es, Fr & De

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Book Description :

Mathematical Proofs: A Transition to Advanced Mathematics, 2/e, prepares students for the more abstract mathematics courses that follow calculus. This text introduces students to proof techniques and writing proofs of their own. As such, it is an introduction to the mathematics enterprise, providing solid introductions to relations, functions, and cardinalities of sets. KEY TOPICS: Communicating Mathematics, Sets, Logic, Direct Proof and Proof by Contrapositive, More on Direct Proof and Proof by Contrapositive, Existence and Proof by Contradiction, Mathematical Induction, Prove or Disprove, Equivalence Relations, Functions, Cardinalities of Sets, Proofs in Number Theory, Proofs in Calculus, Proofs in Group Theory. MARKET: For all readers interested in advanced mathematics and logic.

Understanding Mathematical Proof

Understanding Mathematical Proof Book
Author : John Taylor,Rowan Garnier
Publisher : CRC Press
Release : 2016-04-19
ISBN : 1466514914
Language : En, Es, Fr & De

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Book Description :

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 techniques that mathematicians adopt to prove their results, and offers advice and strategies for constructing proofs. It will improve students’ ability to understand proofs and construct correct proofs of their own. The first chapter of the text introduces the kind of reasoning that mathematicians use when writing their proofs and gives some example proofs to set the scene. The book then describes basic logic to enable an understanding of the structure of both individual mathematical statements and whole mathematical proofs. It also explains the notions of sets and functions and dissects several proofs with a view to exposing some of the underlying features common to most mathematical proofs. The remainder of the book delves further into different types of proof, including direct proof, proof using contrapositive, proof by contradiction, and mathematical induction. The authors also discuss existence and uniqueness proofs and the role of counter examples.

Mathematical Proofs Pearson New International Edition

Mathematical Proofs  Pearson New International Edition Book
Author : Gary Chartrand,Albert D. Polimeni,Ping Zhang
Publisher : Pearson Higher Ed
Release : 2013-10-03
ISBN : 1292052341
Language : En, Es, Fr & De

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Book Description :

Mathematical Proofs: A Transition to Advanced Mathematics, Third Edition, prepares students for the more abstract mathematics courses that follow calculus. Appropriate for self-study or for use in the classroom, this text introduces students to proof techniques, analyzing proofs, and writing proofs of their own. Written in a clear, conversational style, this book provides a solid introduction to such topics as relations, functions, and cardinalities of sets, as well as the theoretical aspects of fields such as number theory, abstract algebra, and group theory. It is also a great reference text that students can look back to when writing or reading proofs in their more advanced courses.

Introduction to Mathematical Proofs

Introduction to Mathematical Proofs Book
Author : Charles Roberts
Publisher : CRC Press
Release : 2009-06-24
ISBN : 9781420069563
Language : En, Es, Fr & De

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Book Description :

Shows How to Read & Write Mathematical Proofs Ideal Foundation for More Advanced Mathematics Courses Introduction 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 necessary to write clear, correct, and concise proofs. Unlike similar textbooks, this one begins with logic since it is the underlying language of mathematics and the basis of reasoned arguments. The text then discusses deductive mathematical systems and the systems of natural numbers, integers, rational numbers, and real numbers. It also covers elementary topics in set theory, explores various properties of relations and functions, and proves several theorems using induction. The final chapters introduce the concept of cardinalities of sets and the concepts and proofs of real analysis and group theory. In the appendix, the author includes some basic guidelines to follow when writing proofs. Written in a conversational style, yet maintaining the proper level of mathematical rigor, this accessible book teaches students to reason logically, read proofs critically, and write valid mathematical proofs. It will prepare them to succeed in more advanced mathematics courses, such as abstract algebra and geometry.

Famous Mathematical Proofs

Famous Mathematical Proofs Book
Author : Edited by Paul F. Kisak
Publisher : Createspace Independent Publishing Platform
Release : 2015-11-20
ISBN : 9781519464330
Language : En, Es, Fr & De

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Book Description :

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms. Proofs are examples of deductive reasoning and are distinguished from inductive or empirical arguments; a proof must demonstrate that a statement is always true (occasionally by listing all possible cases and showing that it holds in each), rather than enumerate many confirmatory cases. An unproved proposition that is believed true is known as a conjecture. Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory. This book contains 'solutions' to some of the most noteworthy mathematical proofs (QED).

An Introduction to Mathematical Proofs

An Introduction to Mathematical Proofs Book
Author : Nicholas A. Loehr
Publisher : CRC Press
Release : 2019
ISBN : 9780429322587
Language : En, Es, Fr & De

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Book Description :

This book contains an introduction to mathematical proofs. The topics appearing in Parts I through VII constitute the standard core material in proofs course. Part VIII develops properties of the real numbers from the ordered field axioms. The book maintains a targeted focus on helping students master key skills needed for later work, as opposed to giving a dry treatment of logic and set theory. A friendly conversational style couples with the necessary level of precision and rigor. The lecture format facilitates a continual cycle of examples, summaries, and review of previous material. Every lecture ends with an immediate review of the main points just covered. Three review lectures give detailed summaries. The essential core material is supplemented by more advanced topics in optional sections. Heavy emphasis is placed on proof templates, creating proof outlines for complex statements based on the logical structure. Many sample proofs are accompanied by annotations. Our coverage of induction is more extensive than some other texts. A careful distinction between the graph of a function and the function itself is made. Key Features: Arranged by fifty one-hour lectures. The lecture format facilitates a continual cycle of examples, summaries, and review of previous material. Parts I and VII cover all the essential topics for a Transition to Advanced Mathematics course. Part VIII offers advanced topics typcially found in an Advanced Calculus course. Heavy emphasis is placed on proof templates, which create proof outlines for complex statements. Induction is covered more than in other texts.

Mathematical Proofs

Mathematical Proofs Book
Author : Source Wikipedia
Publisher : Booksllc.Net
Release : 2013-09
ISBN : 9781230780481
Language : En, Es, Fr & De

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Book Description :

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 58. Chapters: Back-and-forth method, Bijective proof, Cantor's diagonal argument, Combinatorial proof, Commutative diagram, Conditional proof, Constructive proof, Direct proof, Double counting (proof technique), Elementary proof, Equalization (proof), Law of large numbers, List of incomplete proofs, List of long proofs, List of mathematical proofs, Mathematical fallacy, Mathematical induction, Minimal counterexample, Of the form, Original proof of Godel's completeness theorem, Probabilistically checkable proof, Probabilistic method, Probabilistic proofs of non-probabilistic theorems, Proofs from THE BOOK, Proof by contradiction, Proof by contrapositive, Proof by exhaustion, Proof by infinite descent, Proof by intimidation, Proof of impossibility, Proof sketch for Godel's first incompleteness theorem, Proof without words, Q.E.D., Structural induction, Tombstone (typography), Turing's proof. Excerpt: Turing's proof, is a proof by Alan Turing, first published in January 1937 with the title On Computable Numbers, With an Application to the Entscheidungsproblem. It was the second proof of the assertion (Alonzo Church's proof was first) that some decision problems are "undecidable" there is no single algorithm that infallibly gives a correct YES or NO answer to each instance of the problem. In his own words: ..".what I shall prove is quite different from the well-known results of Godel ... I shall now show that there is no general method which tells whether a given formula U is provable in K ..." (Undecidable p. 145). Turing preceded this proof with two others. The second and third both rely on the first. All rely on his development of type-writer-like "computing machines" that obey a simple set of rules and his subsequent development of a "universal computing machine." In 1905 Jules Richard presented this profound paradox. Alan...

The History of Mathematical Proof in Ancient Traditions

The History of Mathematical Proof in Ancient Traditions Book
Author : Karine Chemla
Publisher : Cambridge University Press
Release : 2012-07-05
ISBN : 1139510584
Language : En, Es, Fr & De

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Book Description :

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.

A Transition to Mathematics with Proofs

A Transition to Mathematics with Proofs Book
Author : Michael J Cullinane
Publisher : Jones & Bartlett Publishers
Release : 2011-12-30
ISBN : 1449627781
Language : En, Es, Fr & De

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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.

The Science of Learning Mathematical Proofs

The Science of Learning Mathematical Proofs Book
Author : Elana (St Joseph's College Reiser, Usa),Elana Reiser
Publisher : World Scientific Publishing Company
Release : 2020-11-17
ISBN : 9789811225512
Language : En, Es, Fr & De

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Book Description :

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.

Nonplussed

Nonplussed  Book
Author : Julian Havil
Publisher : Princeton University Press
Release : 2007
ISBN : 9780691120560
Language : En, Es, Fr & De

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Book Description :

Presents a collection of paradoxes from many different areas of math which reveals the math that shows the truth of these and many other unbelievable ideas. This book gives attention to problems from probability and statistics, areas where intuition can easily be wrong. It talks about the history and people associated with many of these problems.

100 Mathematical Proof

100  Mathematical Proof Book
Author : Rowan Garnier,John Taylor
Publisher : John Wiley & Son Limited
Release : 1996-08
ISBN :
Language : En, Es, Fr & De

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Book Description :

"Proof" has been and remains one of the concepts which characterises mathematics. Covering basic propositional and predicate logic as well as discussing axiom systems and formal proofs, the book seeks to explain what mathematicians understand by proofs and how they are communicated. The authors explore the principle techniques of direct and indirect proof including induction, existence and uniqueness proofs, proof by contradiction, constructive and non-constructive proofs, etc. Many examples from analysis and modern algebra are included. The exceptionally clear style and presentation ensures that the book will be useful and enjoyable to those studying and interested in the notion of mathematical "proof."

The Nuts and Bolts of Proofs

The Nuts and Bolts of Proofs Book
Author : Antonella Cupillari
Publisher : Academic Press
Release : 2005-09-08
ISBN : 0080537901
Language : En, Es, Fr & De

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Book Description :

The Nuts and Bolts of Proof instructs students on the basic logic of mathematical proofs, showing how and why proofs of mathematical statements work. It provides them with techniques they can use to gain an inside view of the subject, reach other results, remember results more easily, or rederive them if the results are forgotten.A flow chart graphically demonstrates the basic steps in the construction of any proof and numerous examples illustrate the method and detail necessary to prove various kinds of theorems. * The "List of Symbols" has been extended. * Set Theory section has been strengthened with more examples and exercises. * Addition of "A Collection of Proofs"

Elementary Theory of Metric Spaces

Elementary Theory of Metric Spaces Book
Author : Robert B. Reisel
Publisher : Springer Science & Business Media
Release : 2012-12-06
ISBN : 1461381886
Language : En, Es, Fr & De

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Book Description :

Science students have to spend much of their time learning how to do laboratory work, even if they intend to become theoretical, rather than experimental, scientists. It is important that they understand how experiments are performed and what the results mean. In science the validity of ideas is checked by experiments. If a new idea does not work in the laboratory, it must be discarded. If it does work, it is accepted, at least tentatively. In science, therefore, laboratory experiments are the touchstones for the acceptance or rejection of results. Mathematics is different. This is not to say that experiments are not part of the subject. Numerical calculations and the examina tion of special and simplified cases are important in leading mathematicians to make conjectures, but the acceptance of a conjecture as a theorem only comes when a proof has been constructed. In other words, proofs are to mathematics as laboratory experiments are to science. Mathematics students must, therefore, learn to know what constitute valid proofs and how to construct them. How is this done? Like everything else, by doing. Mathematics students must try to prove results and then have their work criticized by experienced mathematicians. They must critically examine proofs, both correct and incorrect ones, and develop an appreciation of good style. They must, of course, start with easy proofs and build to more complicated ones.

Bridge to Abstract Mathematics

Bridge to Abstract Mathematics Book
Author : Ronald P. Morash
Publisher : McGraw-Hill College
Release : 1991
ISBN :
Language : En, Es, Fr & De

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Book Description :

This text is designed for students who are preparing to take a post-calculus abstract algebra and analysis course. Morash concentrates on providing students with the basic tools (sets, logic and proof techniques) needed for advanced study in mathematics. The first six chapters of the text are devoted to these basics, and these topics are reinforced throughout the remainder of the text. Morash guides students through the transition from a calculus-level courses upper-level courses that have significant abstract mathematical content.

Introduction to Mathematical Structures and Proofs

Introduction to Mathematical Structures and Proofs Book
Author : Larry J. Gerstein
Publisher : Springer Science & Business Media
Release : 2012-06-05
ISBN : 1461442656
Language : En, Es, Fr & De

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Book Description :

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.

Cultural Foundations of Mathematics

Cultural Foundations of Mathematics Book
Author : C. K. Raju
Publisher : Pearson Education India
Release : 2007
ISBN : 9788131708712
Language : En, Es, Fr & De

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Book Description :

The Volume Examines, In Depth, The Implications Of Indian History And Philosophy For Contemporary Mathematics And Science. The Conclusions Challenge Current Formal Mathematics And Its Basis In The Western Dogma That Deduction Is Infallible (Or That It Is Less Fallible Than Induction). The Development Of The Calculus In India, Over A Thousand Years, Is Exhaustively Documented In This Volume, Along With Novel Insights, And Is Related To The Key Sources Of Wealth-Monsoon-Dependent Agriculture And Navigation Required For Overseas Trade - And The Corresponding Requirement Of Timekeeping. Refecting The Usual Double Standard Of Evidence Used To Construct Eurocentric History, A Single, New Standard Of Evidence For Transmissions Is Proposed. Using This, It Is Pointed Out That Jesuits In Cochin, Following The Toledo Model Of Translation, Had Long-Term Opportunity To Transmit Indian Calculus Texts To Europe. The European Navigational Problem Of Determining Latitude, Longitude, And Loxodromes, And The 1582 Gregorian Calendar-Reform, Provided Ample Motivation. The Mathematics In These Earlier Indian Texts Suddenly Starts Appearing In European Works From The Mid-16Th Century Onwards, Providing Compelling Circumstantial Evidence. While The Calculus In India Had Valid Pramana, This Differed From Western Notions Of Proof, And The Indian (Algorismus) Notion Of Number Differed From The European (Abacus) Notion. Hence, Like Their Earlier Difficulties With The Algorismus, Europeans Had Difficulties In Understanding The Calculus, Which, Like Computer Technology, Enhanced The Ability To Calculate, Albeit In A Way Regarded As Epistemologically Insecure. Present-Day Difficulties In Learning Mathematics Are Related, Via Phylogeny Is Ontogeny , To These Historical Difficulties In Assimilating Imported Mathematics. An Appendix Takes Up Further Contemporary Implications Of The New Philosophy Of Mathematics For The Extension Of The Calculus, Which Is Needed To Handle The Infinities Arising In The Study Of Shock Waves And The Renormalization Problem Of Quantum Field Theory.

Math Proofs Demystified

Math Proofs Demystified Book
Author : Stan Gibilisco
Publisher : McGraw Hill Professional
Release : 2005-05-13
ISBN : 0071469923
Language : En, Es, Fr & De

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Book Description :

Almost every student has to study some sort of mathematical proofs, whether it be in geometry, trigonometry, or with higher-level topics. In addition, mathematical theorems have become an interesting course for many students outside of the mathematical arena, purely for the reasoning and logic that is needed to complete them. Therefore, it is not uncommon to have philosophy and law students grappling with proofs. This book is the perfect resource for demystifying the techniques and principles that govern the mathematical proof area, and is done with the standard “Demystified” level, questions and answers, and accessibility.