【2014新书】Understanding Mathematical Proof

Book 图书名称：Understanding Mathematical Proof
Author 作者：Taylor, John; Garnier, Rowan
Publisher 出版社：Taylor & Francis
Page 页数：414
Publishing Date 出版时间：Mar 21, 2014                         
Language 语言：English
Size 大小：2 MB
Format 格式：pdf 文字版
ISBN: 9781466514904, 1466514906
Edition: 第1版  搜索过论坛，没有该文档


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.



== Table of contents ==
Content: Introduction The need for proof The language of mathematics Reasoning Deductive reasoning and truth Example proofs Logic and Reasoning Introduction Propositions, connectives, and truth tables Logical equivalence and logical implication Predicates and quantification Logical reasoning Sets and Functions Introduction Sets and membership Operations on sets The Cartesian product Functions and composite functions Properties of functions The Structure of Mathematical Proofs Introduction Some proofs dissected An informal framework for proofs Direct proof A more formal framework Finding Proofs Direct proof route maps Examples from sets and functions Examples from algebra Examples from analysis Direct Proof: Variations Introduction Proof using the contrapositive Proof of biconditional statements Proof of conjunctions Proof by contradiction Further examples Existence and Uniqueness Introduction Constructive existence proofs Non-constructive existence proofs Counter-examples Uniqueness proofs Mathematical Induction Introduction Proof by induction Variations on proof by induction Hints and Solutions to Selected Exercises Bibliography Index


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