Pure mathematics for beginners : a rigorous introduction to logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra

Pure Mathematics for Beginners Pure Mathematics for Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The 16 lessons in this book cover basic through intermediate material from each of these 8 topics. In addition, all the proofwriting skills that are essential for advanced study in mathematics are covered and reviewed extensively. Pure Mathematics for Beginners is perfect for professors teaching an introductory college course in higher mathematics high school teachers working with advanced math students students wishing to see the type of mathematics they would be exposed to as a math major. The material in this pure math book includes: 16 lessons in 8 subject areas. A problem set after each lesson arranged by difficulty level. A complete solution guide is included as a downloadable PDF file. Pure Math Book Table Of Contents (Selected) Here's a selection from the table of contents:Introduction Lesson 1 - Logic: Statements and Truth Lesson 2 - Set Theory: Sets and Subsets Lesson 3 - Abstract Algebra: Semigroups, Monoids, and Groups Lesson 4 - Number Theory: Ring of Integers Lesson 5 - Real Analysis: The Complete Ordered Field of Reals Lesson 6 - Topology: The Topology of R Lesson 7 - Complex Analysis: The Field of Complex Numbers Lesson 8 - Linear Algebra: Vector Spaces Lesson 9 - Logic: Logical Arguments Lesson 10 - Set Theory: Relations and Functions Lesson 11 - Abstract Algebra: Structures and Homomorphisms Lesson 12 - Number Theory: Primes, GCD, and LCM Lesson 13 - Real Analysis: Limits and Continuity Lesson 14 - Topology: Spaces and Homeomorphisms Lesson 15 - Complex Analysis: Complex Valued Functions Lesson 16 - Linear Algebra: Linear Transformations Cover Legal Notice Title Page Table of Contents Introduction For students For instructors Lesson 1 – Logic: Statements and Truth Statements with Words Statements with Symbols Truth Tables Problem Set 1 Lesson 2 – Set Theory: Sets and Subsets Describing Sets Subsets Unions and Intersections Problem Set 2 Lesson 3 – Abstract Algebra: Semigroups, Monoids, and Groups Binary Operations and Closure Semigroups and Associativity Monoids and Identity Groups and Inverses Problem Set 3 Lesson 4 – Number Theory: The Ring of Integers Rings and Distributivity Divisibility Induction Problem Set 4 Lesson 5 – Real Analysis: The Complete Ordered Field of Reals Fields Ordered Rings and Fields Why Isn’t Q enough? Completeness Problem Set 5 Lesson 6 – Topology: The Topology of R Intervals of Real Numbers Operations on Sets Open and Closed Sets Problem Set 6 Lesson 7 – Complex Analysis: The Field of Complex Numbers A Limitation of the Reals The Complex Field Absolute Value and Distance Basic Topology of C Problem Set 7 Lesson 8 – Linear Algebra: Vector Spaces Vector Spaces Over Fields Subspaces Bases Problem Set 8 Lesson 9 – Logic: Logical Arguments Statements and Substatements Logical Equivalence Validity in Sentential Logic Problem Set 9 Lesson 10 – Set Theory: Relations and Functions Relations Equivalence Relations and Partitions Orderings Functions Equinumerosity Problem Set 10 Lesson 11 – Abstract Algebra: Structures and Homomorphisms Structures and Substructures Homomorphisms Images and Kernels Normal Subgroups and Ring Ideals Problem Set 11 Lesson 12 – Number Theory: Primes, GCD, and LCM Prime Numbers The Division Algorithm GCD and LCM Problem Set 12 Lesson 13 – Real Analysis: Limits and Continuity Strips and Rectangles Limits and Continuity Equivalent Definitions of Limits and Continuity Basic Examples Limit and Continuity Theorems Limits Involving Infinity One-sided Limits Problem Set 13 Lesson 14 – Topology: Spaces and Homeomorphisms Topological Spaces Bases Types of Topological Spaces Continuous Functions and Homeomorphisms Problem Set 14 Lesson 15 – Complex Analysis: Complex Valued Functions The Unit Circle Exponential Form of a Complex Number Functions of a Complex Variable Limits and Continuity The Reimann Sphere Problem Set 15 Lesson 16 – Linear Algebra: Linear Transformations Linear Transformations Matrices The Matrix of a Linear Transformation Images and Kernels Eigenvalues and Eigenvectors Problem Set 16 Index About the Author Books by Dr. Steve Warner "Pure Mathematics for Beginners consists of a series of lessons in logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. This book is ... for any college level course intended to introduce students to proving theorems in higher level mathematics. Due to the diverse amount of content covered, instructors can easily create a wide range of courses by simply choosing from among the 16 lessons in the book. High school and college students that want to begin learning advanced mathematics on their own will also find this book to be quite useful. The book is completely self-contained with no prerequisites. Furthermore, proofs are presented without 'skipping over any steps' and many examples and additional analyses of theorems are presented to help clarify material that many students ordinarily find difficult."--Back cover ## Pure Mathematics for Beginners Pure Mathematics for Beginners consists of a series of lessons in Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra. The 16 lessons in this book cover basic through intermediate material from each of these 8 topics. In addition, all the proofwriting skills that are essential for advanced study in mathematics are covered and reviewed extensively. Pure Mathematics for Beginners is perfect for * professors teaching an introductory college course in higher mathematics * high school teachers working with advanced math students * students wishing to see the type of mathematics they would be exposed to as a math major.