# Pure Mathematics for Beginners

Pure Mathematics for Beginners by Dr. Steve Warner. Contents of Pure Mathematics for Beginners PDF:

• 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.

• 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

Pure Mathematics for Beginners A Rigorous Introduction to Logic, Set Theory, Abstract Algebra, Number Theory, Real Analysis, Topology, Complex Analysis, and Linear Algebra by Dr. Steve Warner.

This book was written to provide a basic but rigorous introduction to pure mathematics, while exposing students to a wide range of mathematical topics in logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra. For students: There are no prerequisites for this book. The content is completely self-contained. Students with a bit of mathematical knowledge may have an easier time getting through some of the material, but no such knowledge is necessary to read this book.

More important than mathematical knowledge is “mathematical maturity.” Although there is no single agreed upon definition of mathematical maturity, one reasonable way to define it is as “one’s ability to analyze, understand, and communicate mathematics.” A student with a higher level of mathematical maturity will be able to move through this book more quickly than a student with a lower level of mathematical maturity. Whether your level of mathematical maturity is low or high, if you are just starting out in pure mathematics, then you’re in the right place. If you read this book the “right way,” then your level of mathematical maturity will continually be increasing.

This increased level of mathematical maturity will not only help you to succeed in advanced math courses, but it will improve your general problem solving and reasoning skills. This will make it easier to improve your performance in college, in your professional life, and on standardized tests such as the SAT, ACT, GRE, and GMAT. So, what is the “right way” to read this book? Simply reading each lesson from end to end without any further thought and analysis is not the best way to read the book.

You will need to put in some effort to have the best chance of absorbing and retaining the material. When a new theorem is presented, don’t just jump right to the proof and read it. Think about what the theorem is saying. Try to describe it in your own words. Do you believe that it is true? If you do believe it, can you give a convincing argument that it is true? If you do not believe that it is true, try to come up with an example that shows it is false, and then figure out why your example does not contradict the theorem. Pick up a pen or pencil. Draw some pictures, come up with your own examples, and try to write your own proof. You may find that this book goes into more detail than other math books when explaining examples, discussing concepts, and proving theorems. This was done so that any student can read this book, and not just students that are naturally gifted in mathematics. So, it is up to you as the student to try to answer questions before they are answered for you.

When a new definition is given, try to think of your own examples before looking at those presented in the book. And when the book provides an example, do not just accept that it satisfies the given definition. Convince yourself. Prove it. Each lesson is followed by a Problem Set. The problems in each Problem Set have been organized into five levels, Level 1 problems being considered the easiest, and Level 5 problems being considered the most difficult. If you want to get just a small taste of pure mathematics, then you can work on the easier problems. If you want to achieve a deeper understanding of the material, take some time to struggle with the harder problems.

For instructors: This book can be used for a wide range of courses. Although the lessons can be taught in the order presented, they do not need to be. The lessons cycle twice among eight subject areas: logic, set theory, abstract algebra, number theory, real analysis, topology, complex analysis, and linear algebra.