Exploring the infinite : an introduction to proof and analysis / Jennifer Brooks
- Author:
- Brooks, Jennifer (Jennifer Kacmarcik)
- Published:
- Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017]
- Physical Description:
- xv, 283 pages ; 25 cm.
- Series:
- Contents:
- Machine generated contents note: I.Fundamentals of Abstract Mathematics -- 1.Basic Notions -- 1.1.A First Look at Some Familiar Number Systems -- 1.1.1.Integers and natural numbers -- 1.1.2.Rational numbers and real numbers -- 1.2.Inequalities -- 1.3.A First Look at Sets and Functions -- 1.3.1.Sets, elements, and subsets -- 1.3.2.Operations with sets -- 1.3.3.Special subsets of R: intervals -- 1.3.4.Functions -- 1.4.Problems -- 2.Mathematical Induction -- 2.1.First Examples -- 2.1.1.Defining sequences through a formula for the n-th term -- 2.1.2.Defining sequences recursively -- 2.2.First Programs -- 2.3.First Proofs: Mathematical Induction -- 2.4.Strong Induction -- 2.5.The Well-Ordering Principle and Induction -- 2.6.Problems -- 3.Basic Logic and Proof Techniques -- 3.1.Logical Statements and Truth Tables -- 3.1.1.Statements and their negations -- 3.1.2.Combining statements -- 3.1.3.Implications -- 3.2.Quantified Statements and Their Negations -- 3.2.1.Writing implications as quantified statements -- 3.3.Proof Techniques -- 3.3.1.Direct proof -- 3.3.2.Proof by contradiction -- 3.3.3.Proof by contraposition -- 3.3.4.The art of the counterexample -- 3.4.Problems -- 4.Sets, Relations, and Functions -- 4.1.Sets -- 4.2.Relations -- 4.2.1.The definition -- 4.2.2.Order relations -- 4.2.3.Equivalence relations -- 4.3.Functions -- 4.3.1.Images and pre-images -- 4.3.2.Injections, surjections, and bijections -- 4.3.3.Compositions of functions -- 4.3.4.Inverse functions -- 4.4.Problems -- 5.Elementary Discrete Mathematics -- 5.1.Basic Principles of Combinatorics -- 5.1.1.The Addition and Multiplication Principles -- 5.1.2.Permutations and combinations -- 5.1.3.Combinatorial identities -- 5.2.Linear Recurrence Relations -- 5.2.1.An example -- 5.2.2.General results -- 5.3.Analysis of Algorithms -- 5.3.1.Some simple algorithms -- 5.3.2.O, Ω, and θ notation -- 5.3.3.Analysis of the binary search algorithm -- 5.4.Problems -- 6.Number Systems; Algebraic Structures -- 6.1.Representations of Natural Numbers -- 6.1.1.Developing an algorithm to convert a number from base 10 to base 2 -- 6.1.2.Proof of the existence and uniqueness of the base b representation of an element of N -- 6.2.Integers and Divisibility -- 6.3.Modular Arithmetic -- 6.3.1.Definition of congruence and basic properties -- 6.3.2.Congruence classes -- 6.3.3.Operations on congruence classes -- 6.4.The Rational Numbers -- 6.5.Algebraic Structures -- 6.5.1.Binary operations -- 6.5.2.Groups -- 6.5.3.Rings and fields -- 6.6.Problems -- 7.Cardinality -- 7.1.The Definition -- 7.2.Finite Sets Revisited -- 7.3.Countably Infinite Sets -- 7.4.Uncountable Sets -- 7.5.Problems -- II.Foundations of Analysis -- 8.Sequences of Real Numbers -- 8.1.The Limit of a Sequence -- 8.1.1.Numerical and graphical exploration -- 8.1.2.The precise definition of a limit -- 8.2.Properties of Limits -- 8.3.Cauchy Sequences -- 8.3.1.Showing that a sequence is Cauchy -- 8.3.2.Showing that a sequence is divergent -- 8.3.3.Properties of Cauchy sequences -- 8.4.Problems -- 9.A Closer Look at the Real Number System -- 9.1.R as a Complete Ordered Field -- 9.1.1.Completeness -- 9.1.2.Why Q is not complete -- 9.1.3.Algorithms for approximating [√]2 -- 9.2.Construction of R -- 9.2.1.An equivalence relation on Cauchy sequences of rational numbers -- 9.2.2.Operations on R -- 9.2.3.Verifying the field axioms -- 9.2.4.Defining order -- 9.2.5.Sequences of real numbers and completeness -- 9.3.Problems -- 10.Series, Part 1 -- 10.1.Basic Notions -- 10.1.1.Definitions -- 10.1.2.Exploring the sequence of partial sums graphically and numerically -- 10.1.3.Basic properties of convergent series -- 10.1.4.Series that diverge slowly: The harmonic series -- 10.2.Infinite Geometric Series -- 10.3.Tests for Convergence of Series -- 10.4.Representations of Real Numbers -- 10.4.1.Base 10 representation -- 10.4.2.Base 10 representations of rational numbers -- 10.4.3.Representations in other bases -- 10.5.Problems -- 11.The Structure of the Real Line -- 11.1.Basic Notions from Topology -- 11.1.1.Open and closed sets -- 11.1.2.Accumulation points of sets -- 11.2.Compact sets -- 11.2.1.Subsequences and limit points -- 11.2.2.First definition of compactness -- 11.2.3.The Heine--Borel property -- 11.3.A First Glimpse at the Notion of Measure -- 11.3.1.Measuring intervals -- 11.3.2.Measure zero -- 11.3.3.The Cantor set -- 11.4.Problems -- 12.Continuous Functions -- 12.1.Sequential Continuity -- 12.1.1.Exploring sequential continuity graphically and numerically -- 12.1.2.Proving that a function is continuous -- 12.1.3.Proving that a function is discontinuous -- 12.1.4.First results -- 12.2.Related Notions -- 12.2.1.The ε-δ condition -- 12.2.2.Uniform continuity -- 12.2.3.The limit of a function -- 12.3.Important Theorems -- 12.3.1.The Intermediate Value Theorem -- 12.3.2.Developing a root-finding algorithm from the proof of the IVT -- 12.3.3.Continuous functions on compact intervals -- 12.4.Problems -- 13.Differentiation -- 13.1.Definition and First Examples -- 13.2.Differentiation Rules -- 13.3.Applications of the Derivative -- 13.4.Problems -- 14.Series, Part 2 -- 14.1.Absolute and Conditional Convergence -- 14.1.1.The first example -- 14.1.2.Summation by Parts and the Alternating Series Test -- 14.1.3.Basic facts about conditionally convergent series -- 14.2.Rearrangements -- 14.2.1.Rearrangements and non-negative series -- 14.2.2.Using Python to explore the alternating harmonic series -- 14.2.3.A general theorem -- 14.3.Problems -- A.A Very Short Course on Python -- A.1.Getting Started -- A.1.1.Why Python? -- A.1.2.Python versions 2 and 3 -- A.2.Installation and Requirements -- A.2.1.Integrated Development Environments (IDEs) -- A.3.Python Basics -- A.3.1.Exploring in the Python console -- A.3.2.Your first programs -- A.3.3.Good programming practice -- A.3.4.Lists and strings -- A.3.5.If ... else structures and comparison operators -- A.3.6.Loop structures -- A.4.Functions -- A.5.Recursion.
- Subject(s):
- Genre(s):
- ISBN:
- 9781498704496 hardcover
1498704492 hardcover - Note:
- Includes index.
View MARC record | catkey: 19463273