Journey into Mathematics: An Introduction to Proofs

(Dover Books on Mathematics)
Joseph J. Rotman

Dover Publications | 2006 | 256 páginas | djvu | 2,48 Mb

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Descrição: This 3-part treatment begins with the mechanics of writing proofs, proceeds to considerations of the area and circumference of circles, and concludes with examinations of complex numbers and their application, via De Moivre's theorem, to real numbers. "I recommend this as a textbook or supplemental textbook." — Brian Rogers, The Mathematical Association of America. 1998 edition.

The Nuts and Bolts of Proofs: An Introduction to Mathematical Proofs

Third Edition
Antonella Cupillari

Academic Press | 2005 | 192 páginas | pdf |

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Descrição: 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"

How to Prove It: A Structured Approach

Daniel J. Velleman


Cambridge University Press | 2006 | 398 páginas | pdf | 23 Mb

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Conteúdos:
Introduction -- Sentential logic -- 1.1 Deductive reasoning and logical connectives -- 1.2 truth tables -- 1.3 variables and sets -- 1.4 operations on sets -- 1.5 The conditional and biconditional connectives -- Quantificational logic -- 2.1 Quantifiers -- 2.2 Equivalences involving quantifiers -- 2.3 More operations on sets -- Proofs -- 3.1 proof strategies -- 3.2 proofs involving negations and conditionals -- 3.3 Proofs involving quantifiers -- 3.4 Proofs involving conjunctions and biconditionals -- 3.5 Proofs involving disjunctions -- 3.6 Existence and uniqueness proofs -- 3.7 More examples of proofs -- Relations -- 4.1 Ordered pairs and cartesian products -- 4.2 Relations -- 4.3 More about relations -- 4.4 Ordering relations -- 4.5 Closures -- 4.6 Equivalence relations -- Functions -- 5.1 Functions -- 5.2 One-to-one and onto -- 5.3 Inverses of functions -- 5.4 Images and inverse images: a research project -- Mathematical induction -- 6.1 Proof by mathematical induction -- 6.2 More examples -- 6.3 Recursion -- 6.4 Strong induction -- 6.5 Closures again -- Infinite sets -- 7.1 Equinumerous sets -- 7.2 Countable and uncountable sets -- 7.3 The cantor--Schroder--Bernstein theorem -- Appendix 1: Solutions to selected exercises -- Appendix 2: Proof designer -- Suggestions for further reading -- Summary for proof techniques -- Index.

Introduction to Proofs in Mathematics

James Franklin, Albert Daoud

Prentice Hall | 1990 | 280 páginas | DjVu | 1,4 Mb

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online: maths.unsw (em capítulos)

How to Read and Do Proofs: An Introduction to Mathematical Thought Process

Daniel Solow

John Wiley& Sons | 1982 | 172 páginas | Djvu | 1,8 Mb

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Descrição: This straightforward guide describes the main methods used to prove mathematical theorems. Shows how and when to use each technique such as the contrapositive, induction and proof by contradiction. Each method is illustrated by step-by-step examples. The Second Edition features new chapters on nested quantifiers and proof by cases, and the number of exercises has been doubled with answers to odd-numbered exercises provided. This text will be useful as a supplement in mathematics and logic courses. Prerequisite is high-school algebra.

Reading, Writing, and Proving: A Closer Look at Mathematics

(Undergraduate Texts in Mathematics)
Ulrich Daepp, Pamela Gorkin

Springer | 2003 | 408 páginas

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PDF - 12 Mb
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Descrição: This book, which is based on Pólya's method of problem solving, aids students in their transition from calculus (or precalculus) to higher-level mathematics. The book begins by providing a great deal of guidance on how to approach definitions, examples, and theorems in mathematics. It ends by providing projects for independent study. Students will follow Pólya's four step process: learn to understand the problem; devise a plan to solve the problem; carry out that plan; and look back and check what the results told them. Special emphasis is placed on reading carefully and writing well. The authors have included a wide variety of examples, exercises with solutions, problems, and over 40 illustrations, chosen to emphasize these goals. Historical connections are made throughout the text, and students are encouraged to use the rather extensive bibliography to begin making connections of their own. While standard texts in this area prepare students for future courses in algebra, this book also includes chapters on sequences, convergence, and metric spaces for those wanting to bridge the gap between the standard course in calculus and one in analysis.

Teaching and Learning Proof Across the Grades

(Studies in Mathematical Thinking and Learning Series)

Despina A. Stylianou, Maria L. Blanton, Eric J. Knuth

Routledge | 2009 | 408 páginas | PDF | 2,4 Mb

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Descrição: In recent years there has been increased interest in the nature and role of proof in mathematics education; with many mathematics educators advocating that proof should be a central part of the mathematics education of students at all grade levels. This important new collection provides that much-needed forum for mathematics educators to articulate a connected K-16 "story" of proof. Such a story includes understanding how the forms of proof, including the nature of argumentation and justification as well as what counts as proof, evolve chronologically and cognitively and how curricula and instruction can support the development of students’ understanding of proof. Collectively these essays inform educators and researchers at different grade levels about the teaching and learning of proof at each level and, thus, help advance the design of further empirical and theoretical work in this area. By building and extending on existing research and by allowing a variety of voices from the field to be heard, Teaching and Learning Proof Across the Grades not only highlights the main ideas that have recently emerged on proof research, but also defines an agenda for future study

Justifying and Proving in School Mathematics: Student Conceptions and School Data

Celia Hoyles e Lulu Healy

University of London. Institute of Education | 1996 | pdf | 5,10 Mb

on-line: esds.ac.uk

Descrição: In recent years there has been considerable interest in reassessing the role of mathematical proof, influenced by developments in computer technology and an increasing awareness of the role of proof in conveying and illuminating as well as verifying mathematical ideas. Research in mathematics education has shown proof to be an elusive concept for many students. This has been one influence underlying the shift away from formal methods in schools to the more process-orientated approaches now enshrined in the UK National Curriculum Using and Applying Mathematics.
In this project a nationwide survey was conducted to ascertain the current profile of conceptions amongst 15-year-old high-attaining students of the validity of a range of modes of justification in geometry and algebra. Analysis of the survey data informed the design of two teaching experiments in these mathematical domains incorporating computer use and aiming specifically to encourage links between empirical and deductive reasoning. Case studies were constructed to evaluate the influence of these innovations on students' understanding of proving the role of formal mathematical proof.
Both strands of the research contributed to the formulation of recommendations concerning the emphasis on and positioning of mathematical proof in the school curriculum.

Proofs and Refutations: The Logic of Mathematical Discovery

Imre Lakatos, John Worrall, Elie Zahar
Cambridge University Press | 1976 | DjVu | 3,12 MB | 188 páginas

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Referência em: MathEduc

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations.