Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop Spis treści

Transcription

1 Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks. Boca Raton [etc.], cop Spis treści Preface xiii I Fundamentals of Abstract Mathematics 1 1 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 : intervals Functions Problems 15 2 Mathematical Induction First Examples Defining sequences through a formula for the n-th term Defining sequences recursively First Programs First Proofs: Mathematical Induction Strong Induction The Well-Ordering Principle and Induction Problems 29 3 Basic Logic and Proof Techniques Logical Statements and Truth Tables Statements and their negations Combining statements Implications Quantified Statements and Their Negations Writing implications as quantified statements Proof Techniques Direct proof Proof by contradiction Proof by contraposition The art of the counterexample 42

2 3.4 Problems 43 4 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 Problems 56 5 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 O, Ω, and Θ notation Analysis of the binary search algorithm Problems 79 6 Number Systems; Algebraic Structures Representations of Natural Numbers Developing an algorithm to convert a number from base 10 to base Proof of the existence and uniqueness of the base b representation of an element of 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 107

3 6.6 Problems Cardinality The Definition Finite Sets Revisited Countably Infinite Sets Uncountable Sets Problems 121 II Foundations of Analysis Sequences of Real Numbers The Limit of a Sequence Numerical and graphical exploration The precise definition of a limit Properties of Limits Cauchy Sequences Showing that a sequence is Cauchy Showing that a sequence is divergent Properties of Cauchy sequences Problems A Closer Look at the Real Number System Rasa Complete Ordered Field Completeness Why is not complete Algorithms for approximating Construction of An equivalence relation on Cauchy sequences of rational numbers Operations on Verifying the field axioms Defining order Sequences of real numbers and completeness Problems Series, Part Basic Notions Definitions 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 173

4 10.4 Representations of Real Numbers Base 10 representation Base 10 representations of rational numbers Representations in other bases Problems 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 Problems 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 ε-δ 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 Problems Differentiation Definition and First Examples Differentiation Rules Applications of the Derivative Problems Series, Part Absolute and Conditional Convergence The first example 242

5 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 Problems 257 A A Very Short Course on Python 259 A.1 Getting Started 259 A.1.1 Why Python? 259 A.1.2 Python versions 2 and A.2 Installation and Requirements 260 A.2.1 Integrated Development Environments (IDEs) 260 A.3 Python Basics 260 A.3.1 Exploring in the Python console 260 A.3.2 Your first programs 262 A.3.3 Good programming practice 263 A.3.4 Lists and strings 265 A.3.5 if else structures and comparison operators 266 A.3.6 Loop structures 268 A.4 Functions 270 A.5 Recursion 273 Index 275 oprac. BPK