Exploring the Infinite

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