This is not just another algebra book. An entire website supports and extends this text. 400+ web exercises: unlimited, randomly-generated practice and worksheets. The book and website each stand alone as a learning environment; together, they're a dynamic duo. Visit http: //www.onemathematicalcat.org and go to Algebra I: then Geometry, Algebra II, Precalculus, and Calculus. While you're learning algebra, you'll also learn that numbers have lots of different names, and that math is the renaming tool. You'll learn that "x" is to math as "cat" is to English. The original "cat" book (One Mathematical Cat, Please! Ideas for anyone who wants to understand mathematics) is also available on Amazon. If you only need the math language ideas, get the original "cat" book. If you need Algebra too, get this book. The Algebra book has the original cat book embedded in it, so you don't need both! Reviewers and users write: " ... wonderfully written and crafted with a care you rarely see" " ... will do a great service to the mathematical educational world" " ... the need for this book is immense" " ... I found meat, potatoes, and pie on every page. It's all dessert." " ... never seen anything so supportive and affirming and reassuring and inspiring as the way you talk us through topics" "THIS IS GREAT!!!! ... this is helping me get heads above the competition. You rock!!!! OneMathematicalCat drives me wild." There are over 175 web exercises that go directly with this book at: http: //www.onemathematicalcat.org/algebra_book/online_problems/table_of_contents.htm All free. All agreeing perfectly with the text---same order of lessons, same notation, same writing style. Free randomly-generated exercises. Free unlimited worksheets/quizzes. Algebra Pinball. Never again will someone say they don't have enough practice. Bound, printed copies are great. You can highlight, write margin notes, and do exercises right in the book. So, the next time you see "x," think "One Mathematical Cat, Please!" and laugh! Enjoy!

Want a fresh start on math? View it in a totally new way? Then read this little book. You'll learn that numbers have lots of different names, and math is the renaming tool. You'll learn that "x" is to math as "cat" is to English. It's an easy read. If you can add, subtract, multiply and divide with numbers like 1, 2, and 3, then you can read this book. Supplement any math class, 7th grade through college. Use for self-learners, home-schoolers, math lovers, math haters. Reviewers and users write: " . . . wonderfully written and crafted with a care you rarely see" " . . . will do a great service to the mathematical educational world" " . . . the need for this book is immense" " . . . I found meat, potatoes, and pie on every page. It's all dessert." " . . . never seen anything so supportive and affirming and reassuring and inspiring as the way you talk us through topics" "THIS IS GREAT!!!! . . . this is helping me get heads above the competition. You rock!!!! One Mathematical Cat drives me wild." An entire website (http: //www.onemathematicalcat.org) grew from this book: Algebra 1, Geometry, Algebra 2, Precalculus, Calculus. Free randomly-generated exercises. Free unlimited worksheets. Algebra Pinball. All the same style. All the same voice. All building on the ideas in this little book. Bound, printed copies are great for students: they can highlight, write margin notes, and do exercises right in the book. So, the next time you see "x," maybe you'll think "One Mathematical Cat, Please!" and laugh! Enjoy!

The entire book is free, online, at: http: //www.onemathematicalcat.org/Math/Calculus_obj/tableOfContentsCalculus.htm This print version is just a convenience for those of us who like to highlight, write in the margins, and feel the paper between our fingers. There's also a Complete Solution Manual, available in both print and digital versions. It has completely written out answers to all the in-section and end-of-section exercises, and the sample tests. An entire website supports this book: http: //www.onemathematicalcat.org Algebra I and II, Geometry, Precalculus (for when you need to brush up on stuff) Calculus index cards (hold the entire course in the palm of your hand) Please don't hesitate to contact the author, Dr. Carol JVF Burns, with any questions or comments. My email is available from my homepage: http: //www.onemathematicalcat.org

An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.

Tough Test Questions? Missed Lectures? Not Enough Time? Fortunately, there's Schaum's. This all-in-one-package includes more than 1,100 fully solved problems, examples, and practice exercises to sharpen your problem-solving skills. Plus, you will have access to 30 detailed videos featuring Math instructors who explain how to solve the most commonly tested problems--it's just like having your own virtual tutor! You’ll find everything you need to build confidence, skills, and knowledge for the highest score possible. More than 40 million students have trusted Schaum's to help them succeed in the classroom and on exams. Schaum's is the key to faster learning and higher grades in every subject. Each Outline presents all the essential course information in an easy-to-follow, topic-by-topic format. You also get hundreds of examples, solved problems, and practice exercises to test your skills. This Schaum's Outline gives you 1,105 fully solved problems Concise explanations of all calculus concepts Expert tips on using the graphing calculator Fully compatible with your classroom text, Schaum's highlights all the important facts you need to know. Use Schaum's to shorten your study time--and get your best test scores!

“Proofs and Fundamentals: A First Course in Abstract Mathematics” 2nd edition is designed as a "transition" course to introduce undergraduates to the writing of rigorous mathematical proofs, and to such fundamental mathematical ideas as sets, functions, relations, and cardinality. The text serves as a bridge between computational courses such as calculus, and more theoretical, proofs-oriented courses such as linear algebra, abstract algebra and real analysis. This 3-part work carefully balances Proofs, Fundamentals, and Extras. Part 1 presents logic and basic proof techniques; Part 2 thoroughly covers fundamental material such as sets, functions and relations; and Part 3 introduces a variety of extra topics such as groups, combinatorics and sequences. A gentle, friendly style is used, in which motivation and informal discussion play a key role, and yet high standards in rigor and in writing are never compromised. New to the second edition: 1) A new section about the foundations of set theory has been added at the end of the chapter about sets. This section includes a very informal discussion of the Zermelo– Fraenkel Axioms for set theory. We do not make use of these axioms subsequently in the text, but it is valuable for any mathematician to be aware that an axiomatic basis for set theory exists. Also included in this new section is a slightly expanded discussion of the Axiom of Choice, and new discussion of Zorn's Lemma, which is used later in the text. 2) The chapter about the cardinality of sets has been rearranged and expanded. There is a new section at the start of the chapter that summarizes various properties of the set of natural numbers; these properties play important roles subsequently in the chapter. The sections on induction and recursion have been slightly expanded, and have been relocated to an earlier place in the chapter (following the new section), both because they are more concrete than the material found in the other sections of the chapter, and because ideas from the sections on induction and recursion are used in the other sections. Next comes the section on the cardinality of sets (which was originally the first section of the chapter); this section gained proofs of the Schroeder–Bernstein theorem and the Trichotomy Law for Sets, and lost most of the material about finite and countable sets, which has now been moved to a new section devoted to those two types of sets. The chapter concludes with the section on the cardinality of the number systems. 3) The chapter on the construction of the natural numbers, integers and rational numbers from the Peano Postulates was removed entirely. That material was originally included to provide the needed background about the number systems, particularly for the discussion of the cardinality of sets, but it was always somewhat out of place given the level and scope of this text. The background material about the natural numbers needed for the cardinality of sets has now been summarized in a new section at the start of that chapter, making the chapter both self-contained and more accessible than it previously was. 4) The section on families of sets has been thoroughly revised, with the focus being on families of sets in general, not necessarily thought of as indexed. 5) A new section about the convergence of sequences has been added to the chapter on selected topics. This new section, which treats a topic from real analysis, adds some diversity to the chapter, which had hitherto contained selected topics of only an algebraic or combinatorial nature. 6) A new section called ``You Are the Professor'' has been added to the end of the last chapter. This new section, which includes a number of attempted proofs taken from actual homework exercises submitted by students, offers the reader the opportunity to solidify her facility for writing proofs by critiquing these submissions as if she were the instructor for the course. 7) All known errors have been corrected. 8) Many minor adjustments of wording have been made throughout the text, with the hope of improving the exposition.

Finally a self-contained, one volume, graduate-level algebra text that is readable by the average graduate student and flexible enough to accommodate a wide variety of instructors and course contents. The guiding principle throughout is that the material should be presented as general as possible, consistent with good pedagogy. Therefore it stresses clarity rather than brevity and contains an extraordinarily large number of illustrative exercises.

New edition includes extensive revisions of the material on finite groups and Galois Theory. New problems added throughout.

From the winner of the Turing Award and the Abel Prize, an introduction to computational complexity theory, its connections and interactions with mathematics, and its central role in the natural and social sciences, technology, and philosophy Mathematics and Computation provides a broad, conceptual overview of computational complexity theory—the mathematical study of efficient computation. With important practical applications to computer science and industry, computational complexity theory has evolved into a highly interdisciplinary field, with strong links to most mathematical areas and to a growing number of scientific endeavors. Avi Wigderson takes a sweeping survey of complexity theory, emphasizing the field’s insights and challenges. He explains the ideas and motivations leading to key models, notions, and results. In particular, he looks at algorithms and complexity, computations and proofs, randomness and interaction, quantum and arithmetic computation, and cryptography and learning, all as parts of a cohesive whole with numerous cross-influences. Wigderson illustrates the immense breadth of the field, its beauty and richness, and its diverse and growing interactions with other areas of mathematics. He ends with a comprehensive look at the theory of computation, its methodology and aspirations, and the unique and fundamental ways in which it has shaped and will further shape science, technology, and society. For further reading, an extensive bibliography is provided for all topics covered. Mathematics and Computation is useful for undergraduate and graduate students in mathematics, computer science, and related fields, as well as researchers and teachers in these fields. Many parts require little background, and serve as an invitation to newcomers seeking an introduction to the theory of computation. Comprehensive coverage of computational complexity theory, and beyond High-level, intuitive exposition, which brings conceptual clarity to this central and dynamic scientific discipline Historical accounts of the evolution and motivations of central concepts and models A broad view of the theory of computation's influence on science, technology, and society Extensive bibliography