A Transition to Abstract Mathematics: Learning Mathematical Thinking and Writing

Constructing concise and correct proofs is one of the most challenging aspects of learning to work with advanced mathematics. Meeting this challenge is a defining moment for those considering a career in mathematics or related fields. A Transition to Abstract Mathematics teaches readers to construct proofs and communicate with the precision necessary for working with abstraction. It is based on two premises: composing clear and accurate mathematical arguments is critical in abstract mathematics, and that this skill requires development and support. Abstraction is the destination, not the starting point.

Maddox methodically builds toward a thorough understanding of the proof process, demonstrating and encouraging mathematical thinking along the way. Skillful use of analogy clarifies abstract ideas. Clearly presented methods of mathematical precision provide an understanding of the nature of mathematics and its defining structure.

After mastering the art of the proof process, the reader may pursue two independent paths. The latter parts are purposefully designed to rest on the foundation of the first, and climb quickly into analysis or algebra. Maddox addresses fundamental principles in these two areas, so that readers can apply their mathematical thinking and writing skills to these new concepts. From this exposure, readers experience the beauty of the mathematical landscape and further develop their ability to work with abstract ideas.

Covers the full range of techniques used in proofs, including contrapositive, induction, and proof by contradictionExplains identification of techniques and how they are applied in the specific problemIllustrates how to read written proofs with many step by step examplesIncludes 20% more exercises than the first edition that are integrated into the material instead of end of chapter

Notation and Assumptions

Section I: Foundations of Logic and Proof Writing Ch 1. Logic
Ch 1. Language and Mathematics
Ch 2. Properties of Real Numbers
Ch 3. Sets and Their Properties
Ch 4. Functions

Section II: Basic Principles of Analysis
Ch 5. The Real Numbers
Ch 6. Sequences of Real Numbers
Ch 7. Functions of a Real Variable

Section III: Basic Principles of Algebra
Ch 6. Groups
Ch 7. Rings

Index