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Abstract Algebra: Theory and Applications
Thomas Judson · University of Puget Sound · 2016
This text is intended for a one- or two-semester undergraduate course in abstract algebra. Traditionally, these courses have covered the theoretical aspects of groups, rings, and fields. However, with the development of computing in the last several decades, applications that involve abstract algebra and discrete mathematics have become increasingly important, and many science, engineering, and computer science students are now electing to minor in mathematics. Though theory still occupies a central role in the subject of abstract algebra and no student should go through such a course without a good notion of what a proof is, the importance of applications such as coding theory and cryptography has grown significantly. Until recently most abstract algebra texts included few if any applications. However, one of the major problems in teaching an abstract algebra course is that for many students it is their first encounter with an environment that requires them to do rigorous proofs. Such students often find it hard to see the use of learning to prove theorems and propositions; applied examples help the instructor provide motivation. This text contains more material than can possibly be covered in a single semester. Certainly there is adequate material for a two-semester course, and perhaps more; however, for a one-semester course it would be quite easy to omit selected chapters and still have a useful text. The order of presentation of topics is standard: groups, then rings, and finally fields. Emphasis can be placed either on theory or on applications. A typical one-semester course might cover groups and rings while briefly touching on field theory, using Chapters 1 through 6, 9, 10, 11, 13 (the first part), 16, 17, 18 (the first part), 20, and 21. Parts of these chapters could be deleted and applications substituted according to the interests of the students and the instructor. A two-semester course emphasizing theory might cover Chapters 1 through 6, 9, 10, 11, 13
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A First Course in Linear Algebra
Ken Kuttler · Lyryx · 2023
This text, originally by K. Kuttler, has been redesigned by the Lyryx editorial team as a first course in linear algebra for science and engineering students who have an understanding of basic algebra. All major topics of linear algebra are available in detail, as well as proofs of important theorems. In addition, connections to topics covered in advanced courses are introduced. The text is designed in a modular fashion to maximize flexibility and facilitate adaptation to a given course outline and student profile. Each chapter begins with a list of student learning outcomes, and examples and diagrams are given throughout the text to reinforce ideas and provide guidance on how to approach various problems. Suggested exercises are included at the end of each section, with selected answers at the end of the text. Lyryx develops and supports open texts, with editorial services to adapt the text for each particular course. In addition, Lyryx provides content-specific formative online assessment, a wide variety of supplements, and in-house support available 7 days/week for both students and instructors.
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How We Got from There to Here: A Story of Real Analysis
Robert Rogers; Eugene Boman · Open SUNY · 2014
The typical introductory real analysis text starts with an analysis of the real number system and uses this to develop the definition of a limit, which is then used as a foundation for the definitions encountered thereafter. While this is certainly a reasonable approach from a logical point of view, it is not how the subject evolved, nor is it necessarily the best way to introduce students to the rigorous but highly non-intuitive definitions and proofs found in analysis. This book proposes that an effective way to motivate these definitions is to tell one of the stories (there are many) of the historical development of the subject, from its intuitive beginnings to modern rigor. The definitions and techniques are motivated by the actual difficulties encountered by the intuitive approach and are presented in their historical context. However, this is not a history of analysis book. It is an introductory analysis textbook, presented through the lens of history. As such, it does not simply insert historical snippets to supplement the material. The history is an integral part of the topic, and students are asked to solve problems that occur as they arise in their historical context. This book covers the major topics typically addressed in an introductory undergraduate course in real analysis in their historical order. Written with the student in mind, the book provides guidance for transforming an intuitive understanding into rigorous mathematical arguments. For example, in addition to more traditional problems, major theorems are often stated and a proof is outlined. The student is then asked to fill in the missing details as a homework problem.
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Introductory Statistics
Barbara Illowsky; Susan Dean; Daniel Birmajer; Bryan Blount · OpenStax · 2023
Introductory Statistics 2e provides an engaging, practical, and thorough overview of the core concepts and skills taught in most one-semester statistics courses. The text focuses on diverse applications from a variety of fields and societal contexts, including business, healthcare, sciences, sociology, political science, computing, and several others. The material supports students with conceptual narratives, detailed step-by-step examples, and a wealth of illustrations, as well as collaborative exercises, technology integration problems, and statistics labs. The text assumes some knowledge of intermediate algebra, and includes thousands of problems and exercises that offer instructors and students ample opportunity to explore and reinforce useful statistical skills.
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Linear Algebra
David Cherney; Tom Denton; Andrew Waldon · University of California, Davis · 2016
We believe the entire book can be taught in twenty five 50-minute lectures to a sophomore audience that has been exposed to a one year calculus course. Vector calculus is useful, but not necessary preparation for this book, which attempts to be self-contained. Key concepts are presented multiple times, throughout the book, often first in a more intuitive setting, and then again in a definition, theorem, proof style later on. We do not aim for students to become agile mathematical proof writers, but we do expect them to be able to show and explain why key results hold. We also often use the review exercises to let students discover key results for themselves; before they are presented again in detail later in the book. The book has been written such that instructors can reorder the chapters (using the La- TeX source) in any (reasonable) order and still have a consistent text. We hammer the notions of abstract vectors and linear transformations hard and early, while at the same time giving students the basic matrix skills necessary to perform computations. Gaussian elimination is followed directly by an “exploration chapter” on the simplex algorithm to open students minds to problems beyond standard linear systems ones. Vectors in Rn and general vector spaces are presented back to back so that students are not stranded with the idea that vectors are just ordered lists of numbers. To this end, we also labor the notion of all functions from a set to the real numbers. In the same vein linear transformations and matrices are presented hand in hand. Once students see that a linear map is specified by its action on a limited set of inputs, they can already understand what a basis is. All the while students are studying linear systems and their solution sets, so after matrices determinants are introduced. This material can proceed rapidly since elementary matrices were already introduced with Gaussian elimination. Only then is a careful discussion of spans, linear independen
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Combinatorics Through Guided Discovery
Kenneth Bogart · Kenneth P. Bogart · 2004
This book is an introduction to combinatorial mathematics, also known as combinatorics. The book focuses especially but not exclusively on the part of combinatorics that mathematicians refer to as “counting.” The book consists almost entirely of problems. Some of the problems are designed to lead you to think about a concept, others are designed to help you figure out a concept and state a theorem about it, while still others ask you to prove the theorem. Other problems give you a chance to use a theorem you have proved. From time to time there is a discussion that pulls together some of the things you have learned or introduces a new idea for you to work with. Many of the problems are designed to build up your intuition for how combinatorial mathematics works. There are problems that some people will solve quickly, and there are problems that will take days of thought for everyone. Probably the best way to use this book is to work on a problem until you feel you are not making progress and then go on to the next one. Think about the problem you couldn't get as you do other things. The next chance you get, discuss the problem you are stymied on with other members of the class. Often you will all feel you've hit dead ends, but when you begin comparing notes and listening carefully to each other, you will see more than one approach to the problem and be able to make some progress. In fact, after comparing notes you may realize that there is more than one way to interpret the problem. In this case your first step should be to think together about what the problem is actually asking you to do. You may have learned in school that for every problem you are given, there is a method that has already been taught to you, and you are supposed to figure out which method applies and apply it. That is not the case here. Based on some simplified examples, you will discover the method for yourself. Later on, you may recognize a pattern that suggests you should try to use this metho
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Active Calculus 2.0
Matt Boelkins; David Austin; Steve Schlicker · Grand Valley State University · 2017
Active Calculus is different from most existing calculus texts in at least the following ways: the text is freely readable online in HTML format and is also available for in PDF; in the electronic format, graphics are in full color and there are live links to java applets; version 2.0 now contains WeBWorK exercises in each chapter, which are fully interactive in the HTML format and included in print in the PDF; the text is open source, and interested users can gain access to the original source files on GitHub; the style of the text requires students to be active learners — there are very few worked examples in the text, with there instead being 3-4 activities per section that engage students in connecting ideas, solving problems, and developing understanding of key calculus concepts; each section begins with motivating questions, a brief introduction, and a preview activity, all of which are designed to be read and completed prior to class; following the WeBWorK exercises in each section, there are several challenging problems that require students to connect key ideas and write to communicate their understanding.
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A Gentle Introduction to the Art of Mathematics
Joseph Fields · Southern Connecticut State University · 2015
This book is designed for the transition course between calculus and differential equations and the upper division mathematics courses with an emphasis on proof and abstraction. The book has been used by the author and several other faculty at Southern Connecticut State University. There are nine chapters and more than enough material for a semester course. Student reviews are favorable. It is written in an informal, conversational style with a large number of interesting examples and exercises, so that a student learns to write proofs while working on engaging problems.
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Calculus for the Life Sciences: A Modeling Approach
James Cornette; Ralph Ackerman · A.T. Still University · 2013
Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the 'best' (engineering) course. In our text, mathematical modeling and difference and differential equations lead, closely follow, and extend the elements of calculus. Chapter one introduces mathematical modeling in which students write descriptions of some observed processes and from these descriptions derive first order linear difference equations whose solutions can be compared with the observed data. In chapters in which the derivatives of algebraic, exponential, or trigonometric functions are defined, biologically motivated differential equations and their solutions are included. The chapter on partial derivatives includes a section on the diffusion partial differential equation. There are two chapters on non-linear difference equations and on systems of two difference equations and two chapters on differential equations and on systems of differential equation.
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Precalculus: An Investigation of Functions
David Lippman; Melonie Rasmussen · David Lippman and Melonie Rasmussen · 2017
Precalculus: An Investigation of Functions is a free, open textbook covering a two-quarter pre-calculus sequence including trigonometry. The first portion of the book is an investigation of functions, exploring the graphical behavior of, interpretation of, and solutions to problems involving linear, polynomial, rational, exponential, and logarithmic functions. An emphasis is placed on modeling and interpretation, as well as the important characteristics needed in calculus. The second portion of the book introduces trigonometry. Trig is introduced through an integrated circle/triangle approach. Identities are introduced in the first chapter, and revisited throughout. Likewise, solving is introduced in the second chapter and revisited more extensively in the third chapter. As with the first part of the book, an emphasis is placed on motivating the concepts and on modeling and interpretation. In addition to the paper homework sets, algorithmetically generated online homework is available as part of a complete course shell package, which also includes a sample syllabus, teacher notes with lecture examples, sample quizzes and exams, printable classwork sheets and handouts, and chapter review problems. If you teach in Washington State, you can find the course shell in the WAMAP.org template course list. For those located elsewhere, you can access the course shell at MyOpenMath.com. A self-study version of the online course exercises is also available on MyOpenMath.com for students wanting to learn the material on their own, or who need a refresher.
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Calculus for the Life Sciences: A Modeling Approach
James Cornette; Ralph Ackerman · A.T. Still University · 2011
Our writing is based on three premises. First, life sciences students are motivated by and respond well to actual data related to real life sciences problems. Second, the ultimate goal of calculus in the life sciences primarily involves modeling living systems with difference and differential equations. Understanding the concepts of derivative and integral are crucial, but the ability to compute a large array of derivatives and integrals is of secondary importance. Third, the depth of calculus for life sciences students should be comparable to that of the traditional physics and engineering calculus course; else life sciences students will be short changed and their faculty will advise them to take the 'best' (engineering) course. In our text, mathematical modeling and difference and differential equations lead, closely follow, and extend the elements of calculus. Chapter one introduces mathematical modeling in which students write descriptions of some observed processes and from these descriptions derive first order linear difference equations whose solutions can be compared with the observed data. In chapters in which the derivatives of algebraic, exponential, or trigonometric functions are defined, biologically motivated differential equations and their solutions are included. The chapter on partial derivatives includes a section on the diffusion partial differential equation. There are two chapters on non-linear difference equations and on systems of two difference equations and two chapters on differential equations and on systems of differential equation.
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Elementary Algebra
John Redden · Saylor Foundation · 2011
It is essential to lay a solid foundation in mathematics if a student is to be competitive in today's global market. The importance of algebra, in particular, cannot be overstated, as it is the basis of all mathematical modeling used in applications found in all disciplines. Traditionally, the study of algebra is separated into a two parts, elementary algebra and intermediate algebra. This textbook, Elementary Algebra, is the first part, written in a clear and concise manner, making no assumption of prior algebra experience. It carefully guides students from the basics to the more advanced techniques required to be successful in the next course. This text is, by far, the best elementary algebra textbook offered under a Creative Commons license. It is written in such a way as to maintain maximum flexibility and usability. A modular format was carefully integrated into the design. For example, certain topics, like functions, can be covered or omitted without compromising the overall flow of the text. An introduction of square roots in Chapter 1 is another example that allows for instructors wishing to include the quadratic formula early to do so. Topics such as these are carefully included to enhance the flexibility throughout. This textbook will effectively enable traditional or nontraditional approaches to elementary algebra. This, in addition to robust and diverse exercise sets, provides the base for an excellent individualized textbook instructors can use free of needless edition changes and excessive costs! A few other differences are highlighted below: Equivalent mathematical notation using standard text found on a keyboard A variety of applications and word problems included in most exercise sets Clearly enumerated steps found in context within carefully chosen examples Alternative methods and notation, modularly integrated, where appropriate Video examples available, in context, within the online version of the textbook Robust and diverse exercise sets with di
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Religion in the Law: An Open Access Casebook
Joe Dunman · L. Joe Dunman · 2026
This casebook features over 70 cases from American courts that involve, in some important way, religious belief and action. The book is divided into sections: First Principles, Establishment, Free Exercise, and Special Problems. Each section includes landmark or otherwise influential cases that have influenced American law and religious practice. Most cases come from the U.S. Supreme Court but the lower federal and state courts are also represented.In the contextual introductions to each section and subpart, I have tried to give the reader a basis for understanding how the cases came about and why I chose them for this book. I have tried to minimize editorial comment. I have cited some scholarship where I think it would be helpful, but please do not mistake this as an attempt to produce a comprehensive treatise on the subject of religion in the law. It is a casebook, and a short one, all things considered. At the end of each introductory part is a short “further reading” list. I chose those articles because I found each of them interesting and useful to understanding the topics that precede them. Their selection is not necessarily an endorsement of each author’s arguments, though I do agree with some of them.I designed this casebook specifically for my own use upper level law school seminar called Law & Religion. Class sessions using this book are intended to be student-led, round table talks with the professor acting as discussion prompter and neutral mediator. Generally, 2-5 cases are assigned for each class session. I selected, edited, and arranged the cases to complement each other thematically and chronologically to the best of my ability. Many of the cases include overlapping topics and could fit into multiple categories, so I took some liberties in their arrangement. Your mileage may vary.
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The Story of Contract Law: Implementing the Bargain
Val Ricks · CALI's eLangdell® Press · 2021
This book is a companion volume to Volume I, "The Story of Contract Law: Formation." Volume I introduces students to law study and teaches basic doctrines of contract formation along with formation defenses. This book, Volume II, The Story of Contract Law: Implementing the Bargain, covers the rest of basic contract doctrine, namely, laws that1) determine the content of the bargain (plain meaning, usage and custom, good faith, mistake in transmission, parol evidence, and express and constructive conditions);2) govern the effect of events that occur after formation (impracticability, frustration, failure of consideration, and risk of loss);3) set remedies—rescission, damages, specific performance—available to courts when liability exists; and4) establish the rights of third parties in contracts by assignment or delegation or as third-party beneficiaries. This book includes many classic teaching cases and introduces new ones. The book also includes many problems, most based on actual cases. The book takes especial care with the doctrine of concurrent conditions, a common-law rule adopted in the late 1700s that required doctrinal readjustment across all the law governing contract performance and remedies. This volume also continues several themes from Volume I. Volume II continues to tie rules to contract law’s central structural idea, that of fair exchange. Also, to the extent helpful to student understanding, Volume II explains doctrines in part through their chronological development. The book introduces the doctrines in the order best conducive to students’ understanding contract law as a regulatory whole; for this volume, it is the order in which the doctrines arise in litigation. Finally, where possible, this volume repeats ideas at helpful points and suggests ties between doctrines so that the structural coherence of contract doctrine becomes easier to understand.
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Intellectual Property: Law & the Information Society—Cases and Materials
James Boyle; Jennifer Jenkins · James Boyle, Jennifer Jenkins · 2016
This book is an introduction to intellectual property law, the set of private legal rights that allows individuals and corporations to control intangible creations and marks—from logos to novels to drug formulae—and the exceptions and limitations that define those rights. It focuses on the three graphmain forms of US federal intellectual property—trademark, copyright and patent—but many of the ideas discussed here apply far beyond those legal areas and far beyond the law of the United States. The book is intended to be a textbook for the basic Intellectual Property class, but because it is an open coursebook, which can be freely edited and customized, it is also suitable for an undergraduate class, or for a business, library studies, communications or other graduate school class. Each chapter contains cases and secondary readings and a set of problems or role-playing exercises involving the material. The problems range from a video of the Napster oral argument to counseling clients about search engines and trademarks, applying the First Amendment to digital rights management and copyright or commenting on the Supreme Court's rulings on gene patents.
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Torts: Cases and Contexts
Eric Johnson · CALI's eLangdell® Press · 2016
Plain-spoken and convivial, this casebook makes a deliberate effort to explain the law, rather than to provide a mere compilation of readings and questions. Simple concepts are presented simply. Complex concepts are broken down and accompanied by examples and problems. By being clear and straightforward, the casebook aims to quickly get students to the point where they can navigate regions of gray and build nuanced arguments. The book is written from the conviction that when students stop to puzzle over something, it should be because the law itself puzzles, not because the book obfuscates. Students describe the book as easy to read. A key aim is context, with explanations of how pieces of doctrine fit into the bigger picture. There is also a continual effort to plug doctrine into the real world of practice, getting students to think about litigation strategy and tactics. Another key feature is a high-degree of organization. Doctrine is explained upfront, independent of and before the cases. After the cases, there is no notes-and-questions mishmash. Historical notes, check-your-understanding questions, questions to ponder, and problems are all separately labeled as such. The readings are rich with variety. The classic cases are here, of course. But there are also atypical readings that allow students to see tort law from different perspectives, including an opening statement, a closing argument, administrative-enforcement letters, an excerpt from a novel, and an opinion on tribal law from a Navajo court. Many selections are also startlingly modern, with facts involving texting-and-driving, alcoholic energy drinks, Facebook libel, suddenly accelerating Toyotas, and the misery of a six-hour tarmac delay. Please note that the publisher requires you to login to access and download the textbooks.
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U.S. Federal Income Taxation of Individuals 2023
Deborah Geier · CALI's eLangdell® Press · 2017
This is the tenth edition of this textbook, updated through December 15, 2022 for use beginning January 2023. In addition to incorporating new law and all inflation adjustments, this edition builds on updates made for the ninth edition and incorporates new charts pertaining to economic and tax data, including December 2019 CBO charts showing that income inequality between 2015 and 2020 (with the Tax Cuts and Jobs Act enacted in 2017) worsened after taxes and transfers than before taxes and transfers are taken into account. This textbook is not intended to be an exhaustive treatise; rather, it is intended to be far more useful than that for beginning tax law students by equipping the novice not merely with unmoored detail but rather with a rich blueprint that illuminates the deeper structural framework on which that detail hangs (sometimes crookedly). Chapter 1 outlines the conceptual meaning of the term income for uniquely tax purposes and examines the Internal Revenue Code provisions that translate this construct into positive law. Chapter 2 explores various forms of consumption taxation because the modern Internal Revenue Code is best perceived as a hybrid income-consumption tax that also contains many provisions that are inconsistent with both forms of taxation. Chapter 3 then provides students with the story of how we got to where we are today, important context about the distribution of the tax burden, the budget, and economic trends, as well as material on ethical debates, economic theories, and politics as they affect taxation. Armed with this larger blueprint, students are in a much better position to evaluate the myriad pieces that follow throughout the remaining 18 chapters. For example, they are in a better position to appreciate how applying the income tax rules for debt to a debt-financed investment afforded more favorable consumption tax treatment creates tax shelter problems. Stated another way, they can better appreciate how the tax system can someti
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Torts: Cases and Contexts
Eric Johnson · CALI's eLangdell® Press · 2016
Plain-spoken and convivial, this casebook makes a deliberate effort to explain the law, rather than to provide a mere compilation of readings and questions. Simple concepts are presented simply. Complex concepts are broken down and accompanied by examples and problems. By being clear and straightforward, the casebook aims to quickly get students to the point where they can navigate regions of gray and build nuanced arguments. The book is written from the conviction that when students stop to puzzle over something, it should be because the law itself puzzles, not because the book obfuscates. Students describe the book as easy to read. A key aim is context, with explanations of how pieces of doctrine fit into the bigger picture. There is also a continual effort to plug doctrine into the real world of practice, getting students to think about litigation strategy and tactics. Another key feature is a high-degree of organization. Doctrine is explained upfront, independent of and before the cases. After the cases, there is no notes-and-questions mishmash. Historical notes, check-your-understanding questions, questions to ponder, and problems are all separately labeled as such. The readings are rich with variety. The classic cases are here, of course. But there are also atypical readings that allow students to see tort law from different perspectives, including an opening statement, a closing argument, administrative-enforcement letters, an excerpt from a novel, and an opinion on tribal law from a Navajo court. Many selections are also startlingly modern, with facts involving texting-and-driving, alcoholic energy drinks, Facebook libel, suddenly accelerating Toyotas, and the misery of a six-hour tarmac delay.
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Sales and Leases: A Problem-based Approach
Scott Burnham; Kristen Juras · CALI's eLangdell® Press · 2023
Sales and Leases is a coursebook for a 3-credit course in personal property sales and leases – the subject matter of UCC Articles 2 and 2A. Adjustments could be made for other credit allocations and chapters can be used on a stand-alone basis. The course is designed so that students both review the rules and principles they studied in their first-year course in Contracts and learn the rules that apply to the subset of contracts for the sale and lease of goods. Students taking this course should be well-prepared to solve legal problems in contracts and sales, and should be well-prepared for those parts of the bar exam as well. While the course in Contracts focuses on the rules of common-law contracts, the focus of this course is the rules found in legislation. Therefore, instead of emphasizing case analysis, the book contains a good deal of narrative that assists students in working through the complexity of the statutes. Students will need to supplement the book with a complete copy of the UCC that includes the Official Comments. Discussion of other statutes, such as UETA and Magnuson-Moss, is incorporated where appropriate. The approach is problem-based, which we believe is more appropriate for an upper-division course based primarily on statutes. The narrative is interspersed with problems for class discussion that require students to apply the principles and rules to particular fact situations. Many times there is an issue of interpretation or policy in the Code, however, and therefore each chapter also contains at least one case that explores an issue arising under the statute.
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Law of Wills
Browne Lewis · CALI's eLangdell® Press · 2016
The purpose of this casebook is to train law students to think and act like probate attorneys. This book is meant to be used in conjunction with the author's book on the law of trusts. This book's focus is problem-solving and legal application; the book includes numerous problems, so law students can learn to apply the law they learn from reading the cases. It also contains collaborative learning exercises to encourage students to engage in group problem-solving. The book is divided into three parts to reflect the main types of issues that students will encounter if they practice probate law. The book's organization mirrors the manner in which probate law is practiced in the real world. The book starts with an examination of the intestacy system because the majority of people die without executing a will. Therefore, most of the legal issues a probate lawyer faces center around the intestacy system. Unlike the typical wills casebook, this book provides a detailed discussion of the intestacy system. A chapter on ethics is included because probate attorneys encounter ethical issues that are different from attorneys practicing in other areas of law. The second part of the book includes an exploration of the testacy system. It is arranged so professors can lead students from the client interview to the will execution. The first three chapters of this section deal with issues that directly impact the existence of the inheritance system. It analyzes a person's ability to control the disposition of his or her property after death. This serves as the students' first introduction to the power of the “dead hand”. These chapters are included to start a public policy discussion about the rights of the dead, the right of heirs, and the necessity of an inheritance system. I tell my students that, when executing a will, they must think of the ways that it can be contested. In addition, I tell them that a will can be contested on two fronts-an attack on the testator and an attack on