Exploring the infinite : an introduction to proof and analysis

著者

• Brooks, Jennifer

書誌事項

Exploring the infinite : an introduction to proof and analysis

Jennifer Brooks

（Textbooks in mathematics）

CRC Press, c2017

注記

Includes index

内容説明・目次

内容説明

Exploring the Infinite addresses the trend toward a combined transition course and introduction to analysis course. It guides the reader through the processes of abstraction and log- ical argumentation, to make the transition from student of mathematics to practitioner of mathematics. This requires more than knowledge of the definitions of mathematical structures, elementary logic, and standard proof techniques. The student focused on only these will develop little more than the ability to identify a number of proof templates and to apply them in predictable ways to standard problems. This book aims to do something more; it aims to help readers learn to explore mathematical situations, to make conjectures, and only then to apply methods of proof. Practitioners of mathematics must do all of these things. The chapters of this text are divided into two parts. Part I serves as an introduction to proof and abstract mathematics and aims to prepare the reader for advanced course work in all areas of mathematics. It thus includes all the standard material from a transition to proof" course. Part II constitutes an introduction to the basic concepts of analysis, including limits of sequences of real numbers and of functions, infinite series, the structure of the real line, and continuous functions. Features Two part text for the combined transition and analysis course New approach focuses on exploration and creative thought Emphasizes the limit and sequences Introduces programming skills to explore concepts in analysis Emphasis in on developing mathematical thought Exploration problems expand more traditional exercise sets

目次

Fundamentals of Abstract Mathematics Basic Notions A First Look at Some Familiar Number Systems Integers and natural numbers Rational numbers and real numbers Inequalities A First Look at Sets and Functions Sets, elements, and subsets Operations with sets Special subsets of R: intervals Functions Mathematical Induction First Examples Defining sequences through a formula for the n-th term Defining sequences recursively First Programs First Proofs: The Principle of Mathematical Induction Strong Induction The Well-Ordering Principle and Induction Basic Logic and Proof Techniques Logical Statements and Truth Table Statements and their negations Combining statements Implications Quantified Statements and Their Negations Writing implications as quanti ed statements Proof Techniques Direct Proof Proof by contradiction Proof by contraposition The art of the counterexample Sets, Relations, and Functions Sets Relations The definition Order Relations Equivalence Relations Functions Images and pre-images Injections, surjections, and bijections Compositions of functions Inverse Functions Elementary Discrete Mathematics Basic Principles of Combinatorics The Addition and Multiplication Principles Permutations and combinations Combinatorial identities Linear Recurrence Relations An example General results Analysis of Algorithms Some simple algorithms Omicron, Omega and Theta notation Analysis of the binary search algorithm Number Systems and Algebraic Structures Representations of Natural Numbers Developing an algorithm to convert a number from base 10 to base 2. Proof of the existence and uniqueness of the base b representation of an element of N Integers and Divisibility Modular Arithmetic Definition of congruence and basic properties Congruence classes Operations on congruence classes The Rational Numbers Algebraic Structures Binary Operations Groups Rings and fields Cardinality The Definition Finite Sets Revisited Countably Infinite Sets Uncountable Sets Foundations of Analysis Sequences of Real Numbers The Limit of a Sequence Numerical and graphical exploration The precise de nition of a limit Properties of Limits Cauchy Sequences Showing that a sequence is Cauchy Showing that a sequence is divergent Properties of Cauchy sequences A Closer Look at the Real Number System R as a Complete Ordered Field Completeness Why Q is not complete Algorithms for approximating square root 2 Construction of R An equivalence relation on Cauchy sequences of rational numbers Operations on R Verifying the field axioms Defining order Sequences of real numbers and completeness Series, Part 1 Basic Notions Exploring the sequence of partial sums graphically and numerically Basic properties of convergent series Series that diverge slowly: The harmonic series Infinite geometric series Tests for Convergence of Series Representations of real numbers Base 10 representation Base 10 representations of rational numbers Representations in other bases The Structure of the Real Line Basic Notions from Topology Open and closed sets Accumulation points of sets Compact sets Subsequences and limit points First definition of compactness The Heine-Borel Property A First Glimpse at the Notion of Measure Measuring intervals Measure zero The Cantor set Continuous Functions Sequential Continuity Exploring sequential continuity graphically and numerically Proving that a function is continuous Proving that a function is discontinuous First results Related Notions The epsilon-delta condition Uniform continuity The limit of a function Important Theorems The Intermediate Value Theorem Developing a root-finding algorithm from the proof of the IVT Continuous functions on compact intervals Differentiation Definition and First Examples Properties of Differentiable Functions and Rules for Differentiation Applications of the Derivative Series, Part 2 Absolutely and Conditionally Convergent Series The rst example Summation by Parts and the Alternating Series Test Basic facts about conditionally convergent series Rearrangements Rearrangements and non-negative series Using Python to explore the alternating harmonic series A general theorem A Very Short Course on Python Getting Stated Why Python? Python versions 2 and 3 Installation and Requirements Integrated Development Environments (IDEs) Python Basics Exploring in the Python Console Your First Programs Good Programming Practice Lists and strings if . . . else structures and comparison operators Loop structures Functions Recursion

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詳細情報

• NII書誌ID(NCID): BB22925674
• ISBN:
  • 9781498704496
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Boca Raton
• ページ数/冊数: xv, 283 p.
• 大きさ: 24 cm
• 分類:
  • LCC : QA300
  • DC23 : 515
• 件名:
  • LCSH : Mathematical analysis -- Textbooks
• 親書誌ID:
  • BA83550953