1st Edition
Exploring the Infinite An Introduction to Proof and Analysis
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
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
Biography
Jennifer Halfpap is an Associate Professor in the Department of Mathematical Sciences at the University of Montana, Missoula, USA. She is also the Associate Chair of the department, directing the Graduate Program.
