An Invitation to Abstract Mathematics

Description

This undergraduate textbook promotes an active transition to higher mathematics. Problem solving is the heart and soul of this book: each problem is carefully chosen to demonstrate, elucidate, or extend a concept. More than 300 exercises engage the reader in extensive arguments and creative approaches, while exploring connections between fundamental mathematical topics. Divided into four parts, this book begins with a playful exploration of the building blocks of mathematics, such as definitions, axioms, and proofs. A study of the fundamental concepts of logic, sets, and functions follows, before focus turns to methods of proof. Having covered the core of a transition course, the author goes on to present a selection of advanced topics that offer opportunities for extension or further study. Throughout, appendices touch on historical perspectives, current trends, and open questions, showing mathematics as a vibrant and dynamic human enterprise. This second edition has been reorganized to better reflect the layout and curriculum of standard transition courses. It also features recent developments and improved appendices. An Invitation to Abstract Mathematics is ideal for those seeking a challenging and engaging transition to advanced mathematics, and will appeal to both undergraduates majoring in mathematics, as well as non-math majors interested in exploring higher-level concepts. From reviews of the first edition: Bajnok’s new book truly invites students to enjoy the beauty, power, and challenge of abstract mathematics. … The book can be used as a text for traditional transition or structure courses … but since Bajnok invites all students, not just mathematics majors, to enjoy the subject, he assumes very little background knowledge. Jill Dietz, MAA Reviews The style of writing is careful, but joyously enthusiastic…. The author’s clear attitude is that mathematics consists of problem solving, and that writing a proof falls into this category. Students of mathematics are, therefore, engaged in problem solving, and should be given problems to solve, rather than problems to imitate. The author attributes this approach to his Hungarian background … and encourages students to embrace the challenge in the same way an athlete engages in vigorous practice. John Perry, zbMATH Preface to Instructors What Kind of Book Is This? How Can One Teach from This Book? Preface to Students Who Is This Book Written For? How Can One Learn from This Book? Acknowledgments Contents Part I What's Mathematics? 1 Let's Play a Game! Problems 2 What's the Name of the Game? Problems 3 How to Make a Statement Problems 4 What's True in Mathematics? Problems A Ten Famous Conjectures Problems and Assignments B Ten Famous Theorems Problems and Assignments Part II The Foundations of Mathematics 5 Let's Be Logical! Problems 6 Setting Examples Problems 7 Quantifier Mechanics Problems 8 Let's Be Functional! Problems C The Foundations of Set Theory Problems and Assignments Part III How to Prove It 9 Universal Proofs Problems 10 The Domino Effect Problems 11 More Domino Games Problems 12 Existential Proofs Problems D Ten Famous Problems Problem 1: Nobody Knows Nothing! Problem 2: Mathematical Survivor Problem 3: From the White House Problem 4: The Towers of Hanoi Problem 5: Checkerboard Tilings and Primino Numbers Problem 6: Watching Your Weights Problem 7: Some Are Sums Problem 8: A Really ``Mean'' Problem Problem 9: Waring's Problem with Negativity Problem 10: Coloring Lines, Planes, and Space Part IV Advanced Math for Beginners 13 Mathematical Structures Problems 14 Working in the Fields (and Other Structures) Problems 15 Group Work Problems 16 Good Relations Problems 17 Order, Please! Problems 18 Now That's the Limit! Problems 19 Sizing It Up Problems 20 Infinite Delights Problems 21 Number Systems Systematically Problems 22 Games Are Valuable! Problems E Graphic Content Problems and Assignments F All Games Considered Problems and Assignments G A Top Forty List of Math Theorems Index