Proofs and Fundamentals PDF Download

Author: Ethan D. Bloch
Publisher: Springer Science & Business Media
ISBN: 1461221307
Category : Mathematics
Languages : en
Pages : 434

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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.

Author: Martin Aigner
Publisher: Springer Science & Business Media
ISBN: 3662223430
Category : Mathematics
Languages : en
Pages : 194

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Book Description
According to the great mathematician Paul Erdös, God maintains perfect mathematical proofs in The Book. This book presents the authors candidates for such "perfect proofs," those which contain brilliant ideas, clever connections, and wonderful observations, bringing new insight and surprising perspectives to problems from number theory, geometry, analysis, combinatorics, and graph theory. As a result, this book will be fun reading for anyone with an interest in mathematics.

Author: Charles Matthews
Publisher: Createspace Independent Publishing Platform
ISBN: 9781717176707
Category :
Languages : en
Pages : 166

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Book Description
This mathematics textbook covers the fundamental ideas used in writing proofs. Proof techniques covered include direct proofs, proofs by contrapositive, proofs by contradiction, proofs in set theory, proofs of existentially or universally quantified predicates, proofs by cases, and mathematical induction. Inductive and deductive reasoning are explored. A straightforward approach is taken throughout. Plenty of examples are included and lots of exercises are provided after each brief exposition on the topics at hand. The text begins with a study of symbolic logic, deductive reasoning, and quantifiers. Inductive reasoning and making conjectures are examined next, and once there are some statements to prove, techniques for proving conditional statements, disjunctions, biconditional statements, and quantified predicates are investigated. Terminology and proof techniques in set theory follow with discussions of the pick-a-point method and the algebra of sets. Cartesian products, equivalence relations, orders, and functions are all incorporated. Particular attention is given to injectivity, surjectivity, and cardinality. The text includes an introduction to topology and abstract algebra, with a comparison of topological properties to algebraic properties. This book can be used by itself for an introduction to proofs course or as a supplemental text for students in proof-based mathematics classes. The contents have been rigorously reviewed and tested by instructors and students in classroom settings.

Author: Bernd S. W. Schröder
Publisher: Wiley
ISBN: 9780470551387
Category : Mathematics
Languages : en
Pages : 0

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Book Description
An accessible introduction to abstract mathematics with an emphasis on proof writing Addressing the importance of constructing and understanding mathematical proofs, Fundamentals of Mathematics: An Introduction to Proofs, Logic, Sets, and Numbers introduces key concepts from logic and set theory as well as the fundamental definitions of algebra to prepare readers for further study in the field of mathematics. The author supplies a seamless, hands-on presentation of number systems, utilizing key elements of logic and set theory and encouraging readers to abide by the fundamental rule that you are not allowed to use any results that you have not proved yet. The book begins with a focus on the elements of logic used in everyday mathematical language, exposing readers to standard proof methods and Russell's Paradox. Once this foundation is established, subsequent chapters explore more rigorous mathematical exposition that outlines the requisite elements of Zermelo-Fraenkel set theory and constructs the natural numbers and integers as well as rational, real, and complex numbers in a rigorous, yet accessible manner. Abstraction is introduced as a tool, and special focus is dedicated to concrete, accessible applications, such as public key encryption, that are made possible by abstract ideas. The book concludes with a self-contained proof of Abel's Theorem and an investigation of deeper set theory by introducing the Axiom of Choice, ordinal numbers, and cardinal numbers. Throughout each chapter, proofs are written in much detail with explicit indications that emphasize the main ideas and techniques of proof writing. Exercises at varied levels of mathematical development allow readers to test their understanding of the material, and a related Web site features video presentations for each topic, which can be used along with the book or independently for self-study. Classroom-tested to ensure a fluid and accessible presentation, Fundamentals of Mathematics is an excellent book for mathematics courses on proofs, logic, and set theory at the upper-undergraduate level as well as a supplement for transition courses that prepare students for the rigorous mathematical reasoning of advanced calculus, real analysis, and modern algebra. The book is also a suitable reference for professionals in all areas of mathematics education who are interested in mathematical proofs and the foundation upon which all mathematics is built.

Author: Konstantine Arkoudas
Publisher: MIT Press
ISBN: 0262342502
Category : Computers
Languages : en
Pages : 1223

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Book Description
A textbook that teaches students to read and write proofs using Athena. Proof is the primary vehicle for knowledge generation in mathematics. In computer science, proof has found an additional use: verifying that a particular system (or component, or algorithm) has certain desirable properties. This book teaches students how to read and write proofs using Athena, a freely downloadable computer language. Athena proofs are machine-checkable and written in an intuitive natural-deduction style. The book contains more than 300 exercises, most with full solutions. By putting proofs into practice, it demonstrates the fundamental role of logic and proof in computer science as no other existing text does. Guided by examples and exercises, students are quickly immersed in the most useful high-level proof methods, including equational reasoning, several forms of induction, case analysis, proof by contradiction, and abstraction/specialization. The book includes auxiliary material on SAT and SMT solving, automated theorem proving, and logic programming. The book can be used by upper undergraduate or graduate computer science students with a basic level of programming and mathematical experience. Professional programmers, practitioners of formal methods, and researchers in logic-related branches of computer science will find it a valuable reference.

Author: Richard H. Hammack
Publisher:
ISBN: 9780989472111
Category : Mathematics
Languages : en
Pages : 314

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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.

Author: Daniel J. Velleman
Publisher: Cambridge University Press
ISBN: 0521861241
Category : Mathematics
Languages : en
Pages : 401

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Book Description
This new edition of Daniel J. Velleman's successful textbook contains over 200 new exercises, selected solutions, and an introduction to Proof Designer software.

Author: Ethan D. Bloch
Publisher: Springer Science & Business Media
ISBN: 0387721762
Category : Mathematics
Languages : en
Pages : 577

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Book Description
This text is a rigorous, detailed introduction to real analysis that presents the fundamentals with clear exposition and carefully written definitions, theorems, and proofs. It is organized in a distinctive, flexible way that would make it equally appropriate to undergraduate mathematics majors who want to continue in mathematics, and to future mathematics teachers who want to understand the theory behind calculus. The Real Numbers and Real Analysis will serve as an excellent one-semester text for undergraduates majoring in mathematics, and for students in mathematics education who want a thorough understanding of the theory behind the real number system and calculus.

Author: David C. Kurtz
Publisher: McGraw-Hill Companies
ISBN:
Category : Mathematics
Languages : en
Pages : 216

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Book Description
This text is designed for the average to strong mathematics major taking a course called Transition to Higher Mathematics, Introduction to Proofs, or Fundamentals of Mathematics. It provides a transition to topics covered in advanced mathematics and covers logic, proofs and sets and emphasizes two important mathematical activities - finding examples of objects with specified properties and writing proofs.

Author: William J. LeVeque
Publisher: Courier Corporation
ISBN: 0486141500
Category : Mathematics
Languages : en
Pages : 292

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Book Description
This excellent textbook introduces the basics of number theory, incorporating the language of abstract algebra. A knowledge of such algebraic concepts as group, ring, field, and domain is not assumed, however; all terms are defined and examples are given — making the book self-contained in this respect. The author begins with an introductory chapter on number theory and its early history. Subsequent chapters deal with unique factorization and the GCD, quadratic residues, number-theoretic functions and the distribution of primes, sums of squares, quadratic equations and quadratic fields, diophantine approximation, and more. Included are discussions of topics not always found in introductory texts: factorization and primality of large integers, p-adic numbers, algebraic number fields, Brun's theorem on twin primes, and the transcendence of e, to mention a few. Readers will find a substantial number of well-chosen problems, along with many notes and bibliographical references selected for readability and relevance. Five helpful appendixes — containing such study aids as a factor table, computer-plotted graphs, a table of indices, the Greek alphabet, and a list of symbols — and a bibliography round out this well-written text, which is directed toward undergraduate majors and beginning graduate students in mathematics. No post-calculus prerequisite is assumed. 1977 edition.