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| computer science set theory: Set Theory for Computing Domenico Cantone, Eugenio Omodeo, Alberto Policriti, 2001-06-26 Set Theory for Computing provides a comprehensive account of set-oriented symbolic manipulation methods suitable for automated reasoning. Its main objective is twofold: 1) to provide a flexible formalization for a variety of set languages, and 2) to clarify the semantics of set constructs firmly established in modern specification languages and in the programming practice. Topics include: semantic unification, decision algorithms, modal logics, declarative programming, tableau-based proof techniques, and theory-based theorem proving. The style of presentation is self-contained, rigorous and accurate. Some familiarity with symbolic logic is helpful but not a requirement. This book is a useful resource for all advanced students, professionals, and researchers in computing sciences, artificial intelligence, automated reasoning, logic, and computational mathematics. It will serve to complement their intuitive understanding of set concepts with the ability to master them by symbolic and logically based algorithmic methods and deductive techniques. |
| computer science set theory: Foundations of Computing Thierry Scheurer, 1994 Written for professionals learning the field of discrete mathematics, this book provides the necessary foundations of computer science without requiring excessive mathematical prerequisites. Using a balanced approach of theory and examples, software engineers will find it a refreshing treatment of applications in programming. |
| computer science set theory: Introductory Logic and Sets for Computer Scientists Nimal Nissanke, 1999 This text provides a practical, modern approach to teaching logic and set theory, equipping students with the necessary mathematical understanding and skills required for the mathematical specification of software. It covers all the areas of mathematics that are considered essential to computer science including logic, set theory, modern algebra (group theory), graph theory and combinatorics, whilst taking into account the diverse mathematical background of the students taking the course. In line with current undergraduate curricula this book uses logic extensively, together with set theory, in mathematical specification of software. Languages such as Z and VDM are used for this purpose. Features Particular emphasis is placed on the application of logic in the fields of software engineering, artificial intelligence and natural language processing 0201179571B04062001 |
| computer science set theory: Computational Logic and Set Theory Jacob T. Schwartz, Domenico Cantone, Eugenio G. Omodeo, 2011-07-16 This must-read text presents the pioneering work of the late Professor Jacob (Jack) T. Schwartz on computational logic and set theory and its application to proof verification techniques, culminating in the ÆtnaNova system, a prototype computer program designed to verify the correctness of mathematical proofs presented in the language of set theory. Topics and features: describes in depth how a specific first-order theory can be exploited to model and carry out reasoning in branches of computer science and mathematics; presents an unique system for automated proof verification in large-scale software systems; integrates important proof-engineering issues, reflecting the goals of large-scale verifiers; includes an appendix showing formalized proofs of ordinals, of various properties of the transitive closure operation, of finite and transfinite induction principles, and of Zorn’s lemma. |
| computer science set theory: Notes on Set Theory Yiannis Moschovakis, 2013-04-17 What this book is about. The theory of sets is a vibrant, exciting math ematical theory, with its own basic notions, fundamental results and deep open problems, and with significant applications to other mathematical theories. At the same time, axiomatic set theory is often viewed as a foun dation ofmathematics: it is alleged that all mathematical objects are sets, and their properties can be derived from the relatively few and elegant axioms about sets. Nothing so simple-minded can be quite true, but there is little doubt that in standard, current mathematical practice, making a notion precise is essentially synonymous with defining it in set theory. Set theory is the official language of mathematics, just as mathematics is the official language of science. Like most authors of elementary, introductory books about sets, I have tried to do justice to both aspects of the subject. From straight set theory, these Notes cover the basic facts about ab stract sets, including the Axiom of Choice, transfinite recursion, and car dinal and ordinal numbers. Somewhat less common is the inclusion of a chapter on pointsets which focuses on results of interest to analysts and introduces the reader to the Continuum Problem, central to set theory from the very beginning. |
| computer science set theory: Mathematical Foundations of Computer Science Peter A. Fejer, Dan A. Simovici, 2012-12-06 Mathematical Foundations of Computer Science, Volume I is the first of two volumes presenting topics from mathematics (mostly discrete mathematics) which have proven relevant and useful to computer science. This volume treats basic topics, mostly of a set-theoretical nature (sets, functions and relations, partially ordered sets, induction, enumerability, and diagonalization) and illustrates the usefulness of mathematical ideas by presenting applications to computer science. Readers will find useful applications in algorithms, databases, semantics of programming languages, formal languages, theory of computation, and program verification. The material is treated in a straightforward, systematic, and rigorous manner. The volume is organized by mathematical area, making the material easily accessible to the upper-undergraduate students in mathematics as well as in computer science and each chapter contains a large number of exercises. The volume can be used as a textbook, but it will also be useful to researchers and professionals who want a thorough presentation of the mathematical tools they need in a single source. In addition, the book can be used effectively as supplementary reading material in computer science courses, particularly those courses which involve the semantics of programming languages, formal languages and automata, and logic programming. |
| computer science set theory: Sets, Logic and Maths for Computing David Makinson, 2012-02-27 This easy-to-follow textbook introduces the mathematical language, knowledge and problem-solving skills that undergraduates need to study computing. The language is in part qualitative, with concepts such as set, relation, function and recursion/induction; but it is also partly quantitative, with principles of counting and finite probability. Entwined with both are the fundamental notions of logic and their use for representation and proof. Features: teaches finite math as a language for thinking, as much as knowledge and skills to be acquired; uses an intuitive approach with a focus on examples for all general concepts; brings out the interplay between the qualitative and the quantitative in all areas covered, particularly in the treatment of recursion and induction; balances carefully the abstract and concrete, principles and proofs, specific facts and general perspectives; includes highlight boxes that raise common queries and clear confusions; provides numerous exercises, with selected solutions. |
| computer science set theory: Elements of Set Theory Herbert B. Enderton, 1977-05-23 This is an introductory undergraduate textbook in set theory. In mathematics these days, essentially everything is a set. Some knowledge of set theory is necessary part of the background everyone needs for further study of mathematics. It is also possible to study set theory for its own interest--it is a subject with intruiging results anout simple objects. This book starts with material that nobody can do without. There is no end to what can be learned of set theory, but here is a beginning. |
| computer science set theory: Basic Category Theory for Computer Scientists Benjamin C. Pierce, 1991-08-07 Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Category theory is a branch of pure mathematics that is becoming an increasingly important tool in theoretical computer science, especially in programming language semantics, domain theory, and concurrency, where it is already a standard language of discourse. Assuming a minimum of mathematical preparation, Basic Category Theory for Computer Scientists provides a straightforward presentation of the basic constructions and terminology of category theory, including limits, functors, natural transformations, adjoints, and cartesian closed categories. Four case studies illustrate applications of category theory to programming language design, semantics, and the solution of recursive domain equations. A brief literature survey offers suggestions for further study in more advanced texts. Contents Tutorial • Applications • Further Reading |
| computer science set theory: Concise Introduction to Logic and Set Theory Iqbal H. Jebril, Hemen Dutta, Ilwoo Cho, 2021-09-30 This book deals with two important branches of mathematics, namely, logic and set theory. Logic and set theory are closely related and play very crucial roles in the foundation of mathematics, and together produce several results in all of mathematics. The topics of logic and set theory are required in many areas of physical sciences, engineering, and technology. The book offers solved examples and exercises, and provides reasonable details to each topic discussed, for easy understanding. The book is designed for readers from various disciplines where mathematical logic and set theory play a crucial role. The book will be of interested to students and instructors in engineering, mathematics, computer science, and technology. |
| computer science set theory: Building Bridges Martin Grötschel, Gyula O.H. Katona, 2010-05-28 Discrete mathematics and theoretical computer science are closely linked research areas with strong impacts on applications and various other scientific disciplines. Both fields deeply cross fertilize each other. One of the persons who particularly contributed to building bridges between these and many other areas is László Lovász, a scholar whose outstanding scientific work has defined and shaped many research directions in the last 40 years. A number of friends and colleagues, all top authorities in their fields of expertise and all invited plenary speakers at one of two conferences in August 2008 in Hungary, both celebrating Lovász’s 60th birthday, have contributed their latest research papers to this volume. This collection of articles offers an excellent view on the state of combinatorics and related topics and will be of interest for experienced specialists as well as young researchers. |
| computer science set theory: Discrete Mathematics for Computer Science Gary Haggard, John Schlipf, Sue Whitesides, 2006 Master the fundamentals of discrete mathematics with DISCRETE MATHEMATICS FOR COMPUTER SCIENCE with Student Solutions Manual CD-ROM! An increasing number of computer scientists from diverse areas are using discrete mathematical structures to explain concepts and problems and this mathematics text shows you how to express precise ideas in clear mathematical language. Through a wealth of exercises and examples, you will learn how mastering discrete mathematics will help you develop important reasoning skills that will continue to be useful throughout your career. |
| computer science set theory: A Book of Set Theory Charles C Pinter, 2014-07-23 This accessible approach to set theory for upper-level undergraduates poses rigorous but simple arguments. Each definition is accompanied by commentary that motivates and explains new concepts. A historical introduction is followed by discussions of classes and sets, functions, natural and cardinal numbers, the arithmetic of ordinal numbers, and related topics. 1971 edition with new material by the author-- |
| computer science set theory: Computational Logic and Proof Theory Georg Gottlob, Alexander Leitsch, Daniele Mundici, 1993 The Third Kurt G |
| computer science set theory: Discrete Mathematics for Computer Science Jon Pierre Fortney, 2020-12-23 Discrete Mathematics for Computer Science: An Example-Based Introduction is intended for a first- or second-year discrete mathematics course for computer science majors. It covers many important mathematical topics essential for future computer science majors, such as algorithms, number representations, logic, set theory, Boolean algebra, functions, combinatorics, algorithmic complexity, graphs, and trees. Features Designed to be especially useful for courses at the community-college level Ideal as a first- or second-year textbook for computer science majors, or as a general introduction to discrete mathematics Written to be accessible to those with a limited mathematics background, and to aid with the transition to abstract thinking Filled with over 200 worked examples, boxed for easy reference, and over 200 practice problems with answers Contains approximately 40 simple algorithms to aid students in becoming proficient with algorithm control structures and pseudocode Includes an appendix on basic circuit design which provides a real-world motivational example for computer science majors by drawing on multiple topics covered in the book to design a circuit that adds two eight-digit binary numbers Jon Pierre Fortney graduated from the University of Pennsylvania in 1996 with a BA in Mathematics and Actuarial Science and a BSE in Chemical Engineering. Prior to returning to graduate school, he worked as both an environmental engineer and as an actuarial analyst. He graduated from Arizona State University in 2008 with a PhD in Mathematics, specializing in Geometric Mechanics. Since 2012, he has worked at Zayed University in Dubai. This is his second mathematics textbook. |
| computer science set theory: Mathematical Foundations of Computer Science Peter A Fejer, Dan A Simovici, 1990-12-05 |
| computer science set theory: Set Theory and Logic Robert R. Stoll, 2012-05-23 Explores sets and relations, the natural number sequence and its generalization, extension of natural numbers to real numbers, logic, informal axiomatic mathematics, Boolean algebras, informal axiomatic set theory, several algebraic theories, and 1st-order theories. |
| computer science set theory: Set Theory Daniel W. Cunningham, 2016-07-18 Set theory can be considered a unifying theory for mathematics. This book covers the fundamentals of the subject. |
| computer science set theory: Sets, Functions, and Logic Keith Devlin, 2018-10-03 Keith Devlin. You know him. You've read his columns in MAA Online, you've heard him on the radio, and you've seen his popular mathematics books. In between all those activities and his own research, he's been hard at work revising Sets, Functions and Logic, his standard-setting text that has smoothed the road to pure mathematics for legions of undergraduate students. Now in its third edition, Devlin has fully reworked the book to reflect a new generation. The narrative is more lively and less textbook-like. Remarks and asides link the topics presented to the real world of students' experience. The chapter on complex numbers and the discussion of formal symbolic logic are gone in favor of more exercises, and a new introductory chapter on the nature of mathematics--one that motivates readers and sets the stage for the challenges that lie ahead. Students crossing the bridge from calculus to higher mathematics need and deserve all the help they can get. Sets, Functions, and Logic, Third Edition is an affordable little book that all of your transition-course students not only can afford, but will actually read...and enjoy...and learn from. About the Author Dr. Keith Devlin is Executive Director of Stanford University's Center for the Study of Language and Information and a Consulting Professor of Mathematics at Stanford. He has written 23 books, one interactive book on CD-ROM, and over 70 published research articles. He is a Fellow of the American Association for the Advancement of Science, a World Economic Forum Fellow, and a former member of the Mathematical Sciences Education Board of the National Academy of Sciences,. Dr. Devlin is also one of the world's leading popularizers of mathematics. Known as The Math Guy on NPR's Weekend Edition, he is a frequent contributor to other local and national radio and TV shows in the US and Britain, writes a monthly column for the Web journal MAA Online, and regularly writes on mathematics and computers for the British newspaper The Guardian. |
| computer science set theory: Set Theory and the Continuum Problem Raymond M. Smullyan, Melvin Fitting, 2010 A lucid, elegant, and complete survey of set theory, this three-part treatment explores axiomatic set theory, the consistency of the continuum hypothesis, and forcing and independence results. 1996 edition. |
| computer science set theory: Combinatorics and Graph Theory John Harris, Jeffry L. Hirst, Michael Mossinghoff, 2009-04-03 These notes were first used in an introductory course team taught by the authors at Appalachian State University to advanced undergraduates and beginning graduates. The text was written with four pedagogical goals in mind: offer a variety of topics in one course, get to the main themes and tools as efficiently as possible, show the relationships between the different topics, and include recent results to convince students that mathematics is a living discipline. |
| computer science set theory: Mathematical Foundations of Computer Science Bhavanari Satyanarayana, T.V. Pradeep Kumar, Shak Mohiddin Shaw, 2019-08-29 Please note: Taylor & Francis does not sell or distribute the Hardback in India, Pakistan, Nepal, Bhutan, Bangladesh and Sri Lanka |
| computer science set theory: Set Theory of the Continuum Haim Judah, Winfried Just, Hugh Woodin, 2012-12-06 Primarily consisting of talks presented at a workshop at the MSRI during its Logic Year 1989-90, this volume is intended to reflect the whole spectrum of activities in set theory. The first section of the book comprises the invited papers surveying the state of the art in a wide range of topics of set-theoretic research. The second section includes research papers on various aspects of set theory and its relation to algebra and topology. Contributors include: J.Bagaria, T. Bartoszynski, H. Becker, P. Dehornoy, Q. Feng, M. Foreman, M. Gitik, L. Harrington, S. Jackson, H. Judah, W. Just, A.S. Kechris, A. Louveau, S. MacLane, M. Magidor, A.R.D. Mathias, G. Melles, W.J. Mitchell, S. Shelah, R.A. Shore, R.I. Soare, L.J. Stanley, B. Velikovic, H. Woodin. |
| computer science set theory: Set Theory, Logic and Their Limitations Moshe Machover, 1996-05-23 This is an introduction to set theory and logic that starts completely from scratch. The text is accompanied by many methodological remarks and explanations. A rigorous axiomatic presentation of Zermelo-Fraenkel set theory is given, demonstrating how the basic concepts of mathematics have apparently been reduced to set theory. This is followed by a presentation of propositional and first-order logic. Concepts and results of recursion theory are explained in intuitive terms, and the author proves and explains the limitative results of Skolem, Tarski, Church and Gödel (the celebrated incompleteness theorems). For students of mathematics or philosophy this book provides an excellent introduction to logic and set theory. |
| computer science set theory: The Joy of Sets Keith Devlin, 2012-12-06 This text covers the parts of contemporary set theory relevant to other areas of pure mathematics. After a review of naïve set theory, it develops the Zermelo-Fraenkel axioms of the theory before discussing the ordinal and cardinal numbers. It then delves into contemporary set theory, covering such topics as the Borel hierarchy and Lebesgue measure. A final chapter presents an alternative conception of set theory useful in computer science. |
| computer science set theory: Discrete Mathematical Structures for Computer Science Bernard Kolman, Robert C. Busby, 1987 This text has been designed as a complete introduction to discrete mathematics, primarily for computer science majors in either a one or two semester course. The topics addressed are of genuine use in computer science, and are presented in a logically coherent fashion. The material has been organized and interrelated to minimize the mass of definitions and the abstraction of some of the theory. For example, relations and directed graphs are treated as two aspects of the same mathematical idea. Whenever possible each new idea uses previously encountered material, and then developed in such a way that it simplifies the more complex ideas that follow. |
| computer science set theory: Problems and Theorems in Classical Set Theory Peter Komjath, Vilmos Totik, 2006-11-22 This volume contains a variety of problems from classical set theory and represents the first comprehensive collection of such problems. Many of these problems are also related to other fields of mathematics, including algebra, combinatorics, topology and real analysis. Rather than using drill exercises, most problems are challenging and require work, wit, and inspiration. They vary in difficulty, and are organized in such a way that earlier problems help in the solution of later ones. For many of the problems, the authors also trace the history of the problems and then provide proper reference at the end of the solution. |
| computer science set theory: On Sets and Graphs Eugenio G. Omodeo, Alberto Policriti, Alexandru I. Tomescu, 2017-05-11 This treatise presents an integrated perspective on the interplay of set theory and graph theory, providing an extensive selection of examples that highlight how methods from one theory can be used to better solve problems originated in the other. Features: explores the interrelationships between sets and graphs and their applications to finite combinatorics; introduces the fundamental graph-theoretical notions from the standpoint of both set theory and dyadic logic, and presents a discussion on set universes; explains how sets can conveniently model graphs, discussing set graphs and set-theoretic representations of claw-free graphs; investigates when it is convenient to represent sets by graphs, covering counting and encoding problems, the random generation of sets, and the analysis of infinite sets; presents excerpts of formal proofs concerning graphs, whose correctness was verified by means of an automated proof-assistant; contains numerous exercises, examples, definitions, problems and insight panels. |
| computer science set theory: Set Theory and Its Philosophy Michael D. Potter, 2004 A wonderful new book ... Potter has written the best philosophical introduction to set theory on the market - Timothy Bays, Notre Dame Philosophical Reviews. |
| computer science set theory: Set Theory Abhijit Dasgupta, 2013-12-11 What is a number? What is infinity? What is continuity? What is order? Answers to these fundamental questions obtained by late nineteenth-century mathematicians such as Dedekind and Cantor gave birth to set theory. This textbook presents classical set theory in an intuitive but concrete manner. To allow flexibility of topic selection in courses, the book is organized into four relatively independent parts with distinct mathematical flavors. Part I begins with the Dedekind–Peano axioms and ends with the construction of the real numbers. The core Cantor–Dedekind theory of cardinals, orders, and ordinals appears in Part II. Part III focuses on the real continuum. Finally, foundational issues and formal axioms are introduced in Part IV. Each part ends with a postscript chapter discussing topics beyond the scope of the main text, ranging from philosophical remarks to glimpses into landmark results of modern set theory such as the resolution of Lusin's problems on projective sets using determinacy of infinite games and large cardinals. Separating the metamathematical issues into an optional fourth part at the end makes this textbook suitable for students interested in any field of mathematics, not just for those planning to specialize in logic or foundations. There is enough material in the text for a year-long course at the upper-undergraduate level. For shorter one-semester or one-quarter courses, a variety of arrangements of topics are possible. The book will be a useful resource for both experts working in a relevant or adjacent area and beginners wanting to learn set theory via self-study. |
| computer science set theory: Statistical Machine Learning Richard Golden, 2020-06-24 The recent rapid growth in the variety and complexity of new machine learning architectures requires the development of improved methods for designing, analyzing, evaluating, and communicating machine learning technologies. Statistical Machine Learning: A Unified Framework provides students, engineers, and scientists with tools from mathematical statistics and nonlinear optimization theory to become experts in the field of machine learning. In particular, the material in this text directly supports the mathematical analysis and design of old, new, and not-yet-invented nonlinear high-dimensional machine learning algorithms. Features: Unified empirical risk minimization framework supports rigorous mathematical analyses of widely used supervised, unsupervised, and reinforcement machine learning algorithms Matrix calculus methods for supporting machine learning analysis and design applications Explicit conditions for ensuring convergence of adaptive, batch, minibatch, MCEM, and MCMC learning algorithms that minimize both unimodal and multimodal objective functions Explicit conditions for characterizing asymptotic properties of M-estimators and model selection criteria such as AIC and BIC in the presence of possible model misspecification This advanced text is suitable for graduate students or highly motivated undergraduate students in statistics, computer science, electrical engineering, and applied mathematics. The text is self-contained and only assumes knowledge of lower-division linear algebra and upper-division probability theory. Students, professional engineers, and multidisciplinary scientists possessing these minimal prerequisites will find this text challenging yet accessible. About the Author: Richard M. Golden (Ph.D., M.S.E.E., B.S.E.E.) is Professor of Cognitive Science and Participating Faculty Member in Electrical Engineering at the University of Texas at Dallas. Dr. Golden has published articles and given talks at scientific conferences on a wide range of topics in the fields of both statistics and machine learning over the past three decades. His long-term research interests include identifying conditions for the convergence of deterministic and stochastic machine learning algorithms and investigating estimation and inference in the presence of possibly misspecified probability models. |
| computer science set theory: Set-Theoretic Methods in Control Franco Blanchini, Stefano Miani, 2015-07-02 The second edition of this monograph describes the set-theoretic approach for the control and analysis of dynamic systems, both from a theoretical and practical standpoint. This approach is linked to fundamental control problems, such as Lyapunov stability analysis and stabilization, optimal control, control under constraints, persistent disturbance rejection, and uncertain systems analysis and synthesis. Completely self-contained, this book provides a solid foundation of mathematical techniques and applications, extensive references to the relevant literature, and numerous avenues for further theoretical study. All the material from the first edition has been updated to reflect the most recent developments in the field, and a new chapter on switching systems has been added. Each chapter contains examples, case studies, and exercises to allow for a better understanding of theoretical concepts by practical application. The mathematical language is kept to the minimum level necessary for the adequate formulation and statement of the main concepts, yet allowing for a detailed exposition of the numerical algorithms for the solution of the proposed problems. Set-Theoretic Methods in Control will appeal to both researchers and practitioners in control engineering and applied mathematics. It is also well-suited as a textbook for graduate students in these areas. Praise for the First Edition This is an excellent book, full of new ideas and collecting a lot of diverse material related to set-theoretic methods. It can be recommended to a wide control community audience. - B. T. Polyak, Mathematical Reviews This book is an outstanding monograph of a recent research trend in control. It reflects the vast experience of the authors as well as their noticeable contributions to the development of this field...[It] is highly recommended to PhD students and researchers working in control engineering or applied mathematics. The material can also be used for graduate courses in these areas. - Octavian Pastravanu, Zentralblatt MATH |
| computer science set theory: Mathematics for Computer Science Eric Lehman, F. Thomson Leighton, Albert R. Meyer, 2017-03-08 This book covers elementary discrete mathematics for computer science and engineering. It emphasizes mathematical definitions and proofs as well as applicable methods. Topics include formal logic notation, proof methods; induction, well-ordering; sets, relations; elementary graph theory; integer congruences; asymptotic notation and growth of functions; permutations and combinations, counting principles; discrete probability. Further selected topics may also be covered, such as recursive definition and structural induction; state machines and invariants; recurrences; generating functions. |
| computer science set theory: A Basis for Theoretical Computer Science M.A. Arbib, A.J. Kfoury, R.N. Moll, 2012-12-06 Computer science seeks to provide a scientific basis for the study of inform a tion processing, the solution of problems by algorithms, and the design and programming of computers. The last forty years have seen increasing sophistication in the science, in the microelectronics which has made machines of staggering complexity economically feasible, in the advances in programming methodology which allow immense programs to be designed with increasing speed and reduced error, and in the development of mathematical techniques to allow the rigorous specification of program, process, and machine. The present volume is one of a series, The AKM Series in Theoretical Computer Science, designed to make key mathe matical developments in computer science readily accessible to under graduate and beginning graduate students. Specifically, this volume takes readers with little or no mathematical background beyond high school algebra, and gives them a taste of a number of topics in theoretical computer science while laying the mathematical foundation for the later, more detailed, study of such topics as formal language theory, computability theory, programming language semantics, and the study of program verification and correctness. Chapter 1 introduces the basic concepts of set theory, with special emphasis on functions and relations, using a simple algorithm to provide motivation. Chapter 2 presents the notion of inductive proof and gives the reader a good grasp on one of the most important notions of computer science: the recursive definition of functions and data structures. |
| computer science set theory: Fundamentals of Discrete Math for Computer Science Tom Jenkyns, Ben Stephenson, 2012-10-16 This textbook provides an engaging and motivational introduction to traditional topics in discrete mathematics, in a manner specifically designed to appeal to computer science students. The text empowers students to think critically, to be effective problem solvers, to integrate theory and practice, and to recognize the importance of abstraction. Clearly structured and interactive in nature, the book presents detailed walkthroughs of several algorithms, stimulating a conversation with the reader through informal commentary and provocative questions. Features: no university-level background in mathematics required; ideally structured for classroom-use and self-study, with modular chapters following ACM curriculum recommendations; describes mathematical processes in an algorithmic manner; contains examples and exercises throughout the text, and highlights the most important concepts in each section; selects examples that demonstrate a practical use for the concept in question. |
| computer science set theory: Algebraic Set Theory André Joyal, Ieke Moerdijk, 1995-09-14 This book offers a new algebraic approach to set theory. The authors introduce a particular kind of algebra, the Zermelo-Fraenkel algebras, which arise from the familiar axioms of Zermelo-Fraenkel set theory. Furthermore, the authors explicitly construct these algebras using the theory of bisimulations. Their approach is completely constructive, and contains both intuitionistic set theory and topos theory. In particular it provides a uniform description of various constructions of the cumulative hierarchy of sets in forcing models, sheaf models and realizability models. Graduate students and researchers in mathematical logic, category theory and computer science should find this book of great interest, and it should be accessible to anyone with a background in categorical logic. |
| computer science set theory: Notes on Logic and Set Theory P. T. Johnstone, 1987-10-08 A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. |
| computer science set theory: Mathematical Tools for Data Mining Dan A. Simovici, Chaabane Djeraba, 2008-08-15 This volume was born from the experience of the authors as researchers and educators,whichsuggeststhatmanystudentsofdataminingarehandicapped in their research by the lack of a formal, systematic education in its mat- matics. The data mining literature contains many excellent titles that address the needs of users with a variety of interests ranging from decision making to p- tern investigation in biological data. However, these books do not deal with the mathematical tools that are currently needed by data mining researchers and doctoral students. We felt it timely to produce a book that integrates the mathematics of data mining with its applications. We emphasize that this book is about mathematical tools for data mining and not about data mining itself; despite this, a substantial amount of applications of mathematical c- cepts in data mining are presented. The book is intended as a reference for the working data miner. In our opinion, three areas of mathematics are vital for data mining: set theory,includingpartially orderedsetsandcombinatorics;linear algebra,with its many applications in principal component analysis and neural networks; and probability theory, which plays a foundational role in statistics, machine learning and data mining. Thisvolumeisdedicatedtothestudyofset-theoreticalfoundationsofdata mining. Two further volumes are contemplated that will cover linear algebra and probability theory. The ?rst part of this book, dedicated to set theory, begins with a study of functionsandrelations.Applicationsofthesefundamentalconceptstosuch- sues as equivalences and partitions are discussed. Also, we prepare the ground for the following volumes by discussing indicator functions, ?elds and?-?elds, and other concepts. |
| computer science set theory: Mathematical Foundation of Computer Science Y. N. Singh, 2005 The Interesting Feature Of This Book Is Its Organization And Structure. That Consists Of Systematizing Of The Definitions, Methods, And Results That Something Resembling A Theory. Simplicity, Clarity, And Precision Of Mathematical Language Makes Theoretical Topics More Appealing To The Readers Who Are Of Mathematical Or Non-Mathematical Background. For Quick References And Immediate Attentions3⁄4Concepts And Definitions, Methods And Theorems, And Key Notes Are Presented Through Highlighted Points From Beginning To End. Whenever, Necessary And Probable A Visual Approach Of Presentation Is Used. The Amalgamation Of Text And Figures Make Mathematical Rigors Easier To Understand. Each Chapter Begins With The Detailed Contents, Which Are Discussed Inside The Chapter And Conclude With A Summary Of The Material Covered In The Chapter. Summary Provides A Brief Overview Of All The Topics Covered In The Chapter. To Demonstrate The Principles Better, The Applicability Of The Concepts Discussed In Each Topic Are Illustrated By Several Examples Followed By The Practice Sets Or Exercises. |
| computer science set theory: Abstract Set Theory Abraham Adolf Fraenkel, 1968 |
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There are additional exercises or computer projects at the ends of many of the chapters. The computer projects usually require a knowledge of programming. All of these exercises and …
Discrete Mathematics Ii Set Theory For Computer Science
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
Set Theory - Definitions - Oswego
De nition Denotation Operations Special Sets Set Operations that Create New Sets Tuples DeMorgan’s Laws Your Turn Set De nition A Set is a collection of entities (things).
Discrete Mathematics Ii Set Theory For Computer Science …
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
Computer Science Set Theory - origin-biomed.waters
computer science set theory: Notes on Set Theory Yiannis Moschovakis, 2013-04-17 What this book is about. The theory of sets is a vibrant, exciting math ematical theory, with its own basic …
The Theory of Languages and Computation - University of …
1.3 Set Theory Most people are introduced to computer science by using a real computer of course, and for the most part this requires a knowledge of only some basic algebra. But as …
Discrete Mathematics Ii Set Theory For Computer Science
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
A taste of category theory for computer scientists
Category theory, like set theory, is a fundamental for mathematical discourse. In set theory, the most primitive concept is element Sets are built out of elements; relations, functions, and the …
MathematicalLogicinComputerScience - arXiv.org
to computer science, or that a good deal of it has played no role in computer science (e.g., think of the very significant part of set theory that deals with large cardinals).2 Nonetheless, there are …
Mathematics for Computer Science - MIT OpenCourseWare
Mathematics for Computer Science. revised Monday 18. th. May, 2015, 01:43. Eric Lehman. Google Inc. F Thomson Leighton. Department of Mathematics ... 16.5 Set Theory and …
Discrete Mathematics For Computer Science Solution (2024)
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Discrete Mathematics For Computer Science Solution
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Discrete Mathematics For Computer Science Solution
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Na¨ıve Type Theory
School for Computer Science, University of Nottingham, Nottingham, UK, txa@cs.nott.ac.uk Summary. We introduce Type Theory, including Homotopy Type Theory, as an alternative to …
DIGITAL NOTES ON MATHEMATICAL FOUNDATIONS OF …
Graph Theory : Representation of Graph, DFS, BFS, Spanning Trees, planar Graphs. Graph Theory and Applications, Basic Concepts Isomorphism and Sub graphs, Multi graphs and …
Chapter 1 The Foundations Logic And Proof Sets And (2024)
5. Why is understanding set theory important in computer science? Set theory forms the basis for many data structures and algorithms in computer science, including databases, graph theory, …
Discrete Mathematics Ii Set Theory For Computer Science …
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
MIT 6.042J/18.062J Set Theory
Mathematics for Computer Science MIT 6.042J/18.062J Set Theory: ZFC Zermelo-Frankel Set Theory Axioms of Zermelo-Frankel with the Choice axiom (ZFC) define the standard Theory of …
Discrete Mathematics For Computer Science Solution …
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Chapter 1. Introduction to Graph Theory [10pt] (Chapters 1.1, …
V is a set of vertices. It can be any set. E is the set of edges. Each edge has form (u, v) with u, v 2V, u , v. It is permissible to have both (4, 5) and (5, 4), since they are distinct. Prof. Tesler Ch. …
Map theory - hjemmesider.diku.dk
Theoretical Computer Science 102 (1992) l-133 Elsevier Fundamental Study Map theory Klaus Grue DIKU, The Unioersity of Copenhagen, Uniuersitetsparken 1, DK-2100 Copenhagen, …
CS411 & CS675 2015F-01 Set Theory & Proof Techniques
Set Theory & Proof Techniques David Galles Department of Computer Science University of San Francisco. 01-0: Syllabus Office Hours Course Text Prerequisites Test Dates & Testing …
Discrete Mathematics For Computer Science Solution (book)
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Category Theory for Programming - arXiv.org
The scientific literature on category theory in computer science is vast. We list some learning material on category theory. Pierce’s book [9] (available for free) gives a brief introduction to …
MATHEMATICAL FOUNDATIONS OF COMPUTER SCIENCE
COMPUTER SCIENCE Subject Code: CS303ES Regulations : R16 - JNTUH Class : II Year B.Tech CSE I Semester Department of Computer Science and Engineering ... Set theory: …
Discrete Mathematics - (Sets) - Stony Brook University
Operationsonsets Definition LetAandBbesubsetsofauniversalsetU. 1.TheunionofAandB,denotedA∪B,isthesetofallelements thatareinatleastoneofAorB. A∪B= …
Discrete Mathematics Ii Set Theory For Computer Science …
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
Discrete Mathematics II: Set Theory for Computer Science …
ideas in Computer Science from theory to practice; Computer Science, being a science of the arti cial, has had many of its constructs and ideas inspired by Set Theory. The strong tradition, …
Discrete Mathematics For Computer Science Solution (book)
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Discrete Mathematics For Computer Science Solution …
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Mathematics for Computer Science MIT 6.042J/18.062J …
Mathematics for Computer Science MIT 6.042J/18.062J Set Theory: ZFC Albert R Meyer, March 4, 2015 ZF.2 Zermelo-Frankel Set Theory Axioms of Zermelo-Frankel with the Choice axiom …
Discrete Mathematics Ii Set Theory For Computer Science …
computer science topics. This post will equip you with the knowledge and practical strategies to not just survive, but thrive, in your Set Theory journey. Understanding the Foundation: What is …
Enderton Elements Of Set Theory Solutions (book)
Computer Science: Set theory underlies fundamental concepts in computer science, such as data structures, algorithms, and programming languages. Artificial Intelligence: Set theory is …
Discrete Mathematics For Computer Science Solution [PDF]
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
Chapter 1 The Foundations Logic And Proof Sets And (book)
techniques, and set theory, crucial building blocks for advanced mathematics and computer science. We'll explore these topics with clear explanations, step-by-step instructions, practical …
Chapter 1 The Foundations Logic And Proof Sets And (book)
5. Why is understanding set theory important in computer science? Set theory forms the basis for many data structures and algorithms in computer science, including databases, graph theory, …
Discrete Mathematics For Computer Science Solution (PDF)
encountered in this field. From foundational concepts like set theory and logic to advanced topics such as graph theory and algorithms, this resource equips readers with the mathematical tools …
