##### How to Use This Book

While the book is divided into chapters, the main organizational unit is the thirty-seven sections. Each contains a selection of definitions, theorems, and examples interspersed with commentary. If you are enrolled in a course, read the section *before* class and then answer the section's reading questions as preparation for class.

The version available for viewing in a web browser is the most complete, integrating all of the components of the book. Consider acquainting yourself with this version. Knowls are indicated by a dashed underlines and will allow you to seamlessly remind yourself of the content of definitions, theorems, examples, exercises, subsections and more. Use them liberally.

Historically, mathematics texts have numbered definitions and theorems. We have instead adopted a strategy more appropriate to the heavy cross-referencing, linking and knowling afforded by modern media. Mimicking an approach taken by Donald Knuth, we have given items short titles and associated acronyms. You will become comfortable with this scheme after a short time, and might even come to appreciate its inherent advantages. In the web version, each chapter has a list of ten or so important items from that chapter, and you will find yourself recognizing some of these acronyms with no extra effort beyond the normal amount of study. Bruno Mello suggests that some say an acronym should be pronouncable as a word (such as “radar”), and otherwise is an abbreviation. We will not be so strict in our use of the term.

Exercises come in three flavors, indicated by the first letter of their label. “C” indicates a problem that is essentially computational. “T” represents a problem that is more theoretical, usually requiring a solution that is as rigorous as a proof. “M” stands for problems that are “medium”, “moderate”, “midway”, “mediate” or “median”, but never “mediocre.” Their statements could feel computational, but their solutions require a more thorough understanding of the concepts or theory, while perhaps not being as rigorous as a proof. Of course, such a tripartite division will be subject to interpretation. Otherwise, larger numerical values indicate greater perceived difficulty, with gaps allowing for the contribution of new problems from readers. Many, but not all, exercises have complete solutions. These are indicated by daggers in the PDF and print versions, with solutions available in an online supplement, while in the web version a solution is indicated by a knowl right after the problem statement. Resist the urge to peek early. Working the exercises diligently is the best way to master the material.

The Archetypes are a collection of twenty-four archetypical examples. The open source lexical database, WordNet, defines an archetype as “something that serves as a model or a basis for making copies.” We employ the word in the first sense here. By carefully choosing the examples we hope to provide at least one example that is interesting and appropriate for many of the theorems and definitions, and also provide counterexamples to conjectures (and especially counterexamples to converses of theorems). Each archetype has numerous computational results which you could strive to duplicate as you encounter new definitions and theorems. There are some exercises which will help guide you in this quest.