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How to Prove It: A Structured Approach de Daniel J. Velleman

How to Prove It: A Structured Approach

Autor Daniel J. Velleman
en Limba Engleză Paperback – sep 2019

Remarcăm How to Prove It: A Structured Approach ca fiind un instrument fundamental pentru nivelul de licență, facilitând tranziția critică de la rezolvarea algoritmică a problemelor la rigoarea demonstrațiilor teoretice. În această a treia ediție, Daniel J. Velleman rafinează o metodologie deja consacrată prin adăugarea unui capitol esențial de teoria numerelor și extinderea setului de aplicații practice la peste 500 de exerciții. Apreciem în mod deosebit structura logică a volumului: acesta nu se limitează la prezentarea rezultatelor, ci expune mecanismele interne ale gândirii matematice prin secțiuni detaliate de ciornă ('scratch work'), care dezvăluie modul în care se construiesc argumentele despre mulțimi, relații și funcții. Progresia narativă este riguroasă, pornind de la fundamentele logicii sentențiale și cuantificaționale pentru a oferi limbajul necesar abordării inducției matematice și a mulțimilor infinite. Cititorii familiarizați cu Introduction to Mathematical Proofs de Charles Roberts vor aprecia aici abordarea mai structurată și accentul pus pe tehnica de scriere, nu doar pe înțelegerea sistemelor deductive. Față de alte lucrări ale autorului, precum Calculus, care se concentrează pe aplicarea analizei, acest volum acționează ca o punte spre matematica abstractă, similară cu direcția explorată în Philosophies of Mathematics, dar cu un caracter mult mai aplicat. Tonul este pedagogic și accesibil, reușind să demistifice rigoarea axiomatică fără a face compromisuri în privința preciziei academice, ceea ce o face o resursă valoroasă nu doar pentru matematicieni, ci și pentru informaticieni sau logicieni.

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Specificații

ISBN-13: 9781108439534
ISBN-10: 1108439535
Pagini: 468
Ilustrații: 10 tables, 536exercises
Dimensiuni: 151 x 225 x 23 mm
Greutate: 0.67 kg
Ediția:19003Revizuită
Editura: Cambridge University Pr.
Locul publicării:New York, United States

De ce să citești această carte

Această carte este recomandată studenților care doresc să stăpânească arta demonstrației matematice. Prin parcurgerea ei, cititorul câștigă capacitatea de a citi și scrie texte matematice complexe, transformând logica abstractă într-un instrument de lucru concret. Este un ghid practic ce oferă claritate metodologică, fiind ideal pentru cei care se pregătesc pentru cursuri avansate de algebră, analiză sau informatică teoretică.

Despre autor

Daniel J. Velleman este un matematician distins, absolvent summa cum laude al Dartmouth College și doctor al University of Wisconsin-Madison. Cu o carieră academică solidă ce include perioade de predare la University of Texas-Austin, Velleman s-a remarcat prin capacitatea de a explica fundamentele matematice și filosofice ale disciplinei. A fost premiat cu distincții prestigioase precum Lester R. Ford Award și Carl B. Allendoerfer Award pentru contribuțiile sale editoriale. Lucrările sale, de la manuale de calcul la explorări ale filosofiei matematicii, reflectă un angajament constant pentru claritatea expunerii și rigoarea logică.

Descriere scurtă

Proofs play a central role in advanced mathematics and theoretical computer science, yet many students struggle the first time they take a course in which proofs play a significant role. This bestselling text's third edition helps students transition from solving problems to proving theorems by teaching them the techniques needed to read and write proofs. Featuring over 150 new exercises and a new chapter on number theory, this new edition introduces students to the world of advanced mathematics through the mastery of proofs. The book begins with the basic concepts of logic and set theory to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for an analysis of techniques that can be used to build up complex proofs step by step, using detailed 'scratch work' sections to expose the machinery of proofs about numbers, sets, relations, and functions. Assuming no background beyond standard high school mathematics, this book will be useful to anyone interested in logic and proofs: computer scientists, philosophers, linguists, and, of course, mathematicians.

Cuprins

1. Sentential logic; 2. Quantificational logic; 3. Proofs; 4. Relations; 5. Functions; 6. Mathematical induction; 7. Number theory; 8. Infinite sets.

Recenzii

'Not only does this book help students learn how to prove results, it highlights why we care so much. It starts in the introduction with some simple conjectures and gathering data, quickly disproving the first but amassing support for the second. Will that pattern persist? How can these observations lead us to a proof? The book is engagingly written, and covers - in clear and great detail - many proof techniques. There is a wealth of good exercises at various levels. I've taught problem solving before (at The Ohio State University and Williams College), and this book has been a great addition to the resources I recommend to my students.' Steven J. Miller, Williams College, Massachusetts
'This book is my go-to resource for students struggling with how to write mathematical proofs. Beyond its plentiful examples, Velleman clearly lays out the techniques and principles so often glossed over in other texts.' Rafael Frongillo, University of Colorado, Boulder
'I've been using this book religiously for the last eight years. It builds a strong foundation in proof writing and creates the axiomatic framework for future higher-level mathematics courses. Even when teaching more advanced courses, I recommend students to read chapter 3 (Proofs) since it is, in my opinion, the best written exposition of proof writing techniques and strategies. This third edition brings a new chapter (Number Theory), which gives the instructor a few more topics to choose from when teaching a fundamental course in mathematics. I will keep using it and recommending it to everyone, professors and students alike.' Mihai Bailesteanu, Central Connecticut State University
'Professor Velleman sets himself the difficult task of bridging the gap between algorithmic and proof-based mathematics. By focusing on the basic ideas, he succeeded admirably. Many similar books are available, but none are more treasured by beginning students. In the Third Edition, the constant pursuit of excellence is further reinforced.' Taje Ramsamujh, Florida International University
'Proofs are central to mathematical development. They are the tools used by mathematicians to establish and communicate their results. The developing mathematician often learns what constitutes a proof and how to present it by osmosis. How to Prove It aims at changing that. It offers a systematic introduction to the development, structuring, and presentation of logical mathematical arguments, i.e. proofs. The approach is based on the language of first-order logic and supported by proof techniques in the style of natural deduction. The art of proving is exercised with naive set theory and elementary number theory throughout the book. As such, it will prove invaluable to first-year undergraduate students in mathematics and computer science.' Marcelo Fiore, University of Cambridge
'Overall, this is an engagingly-written and effective book for illuminating thinking about and building a careful foundation in proof techniques. I could see it working in an introduction to proof course or a course introducing discrete mathematics topics alongside proof techniques. As a self-study guide, I could see it working as it so well engages the reader, depending on how able they are to navigate the cultural context in some examples.' Peter Rowlett, LMS Newsletter
'Altogether this is an ambitious and largely very successful introduction to the writing of good proofs, laced with many good examples and exercises, and with a pleasantly informal style to make the material attractive and less daunting than the length of the book might suggest. I particularly liked the many discussions of fallacious or incomplete proofs, and the associated challenges to readers to untangle the errors in proofs and to decide for themselves whether a result is true.' Peter Giblin, University of Liverpool, The Mathematical Gazette