Discrete Mathematics – Combinatorics, Counting, Proofs, Recurrences, Solutions (eng)

Discrete Mathematics presents the material in an easily accessible manner. Beside the usual content (expanded a little bit), a special writing style is used.

We start with the natural numbers, function and relations, as well as the powerset lattice. The second chapter illustrates set theory with its laws and the Generalized Continuum Hypothesis. Chapter 3 delivers Boolean Algebra, with the double exponential formula for the # different boolean functions. The next chapter covers the induction, direct, and indirect proof. Chapter 5 presents combinatorics: Rules of sum, product, quotient, the pigeonhole principle, in/exclusion, permutation and binomial coefficient, plus Stirling numbers of 1^st and 2^nd kind as well as the Bell number; additionally, the recurrence relation with back- and forward reasoning is offered. We conclude with general and conditional probability, incl. the Monty Hall problem.

Discrete Mathematics presents the material in a lively fashion, including topics which are usually not presented. Providing an annex with questions and solutions it offers the chance to the readers worldwide to grasp the subject in this handy and clearly arranged treatise.

Walter Hower was born into a farming family. After Kindergarten and primary as well as secondary school he attended the high school – and enjoyed his favourite subject Mathematics. After his military period as air force NATO soldier he studied Computer Science (including the minor Economics) – with the main focus on Artificial Intelligence – at the University of Kaiserslautern, with a really heavy pillar on Mathematics (incl. Mathematik für Informatiker II [Prof. Dr. Heinz Lüneburg]). Already at that time he thought about the need or an alternative to present this difficult material – the inspiration for this book.

Under supervision of Prof. PhD Jörg Siekmann he obtained the university (master) degree "Diplom-Informatiker". Later on, he worked as research assistant under the guidance of Prof. Dr. W. Bibel and as PhD candidate in the group of Prof. Dr. Manfred Rosendahl, where he obtained the degree "Dr. rer. nat." for his doctoral thesis "On constraint satisfaction and computer-aided layout design". Immediately afterwards, Dr. Hower accepted the offer by Prof. PhD Jim Bowen to head a research group at University College Cork, National University of Ireland, where he invigorates an expert team on Constraint Processing – which finally evolved into the famous Cork Constraint Computation Centre. After a period in research labs and business companies, since more than 20 years the author now works as professor for Computer Science and Fundamentals of Mathematics; he can be reached as follows:

Albstadt-Sigmaringen University, Fakultät Informatik, Poststr. 6, D-72458 Albstadt-Ebingen, hower@hs-albsig.de.

Prof. Dr. Hower received the Teaching Prize 2006 of the state Baden-Württemberg and germany-wide the 3^rd place Professor of the Year in Engineering / Computer Science in 2009.

Scientific activities include:

- An Approach to Constraints, Interner Bericht IITB/TR-86-7,Forschungsgruppe Expertensysteme, Fraunhofer-Institut fürInformations- und Datenverarbeitung, Karlsruhe, 1986/1987

- A Lattice-based Constraint Formalism, First Australian KnowledgeEngineering Congress, Workshop on AI & Creativity, Melbourne, Australia,March, 1989

- On Conflict Resolution in Inconsistent Constraint Networks,Diplomarbeit, Fachbereich Informatik, Universität Kaiserslautern, 1989,Forschungsbericht AIDA-90-02, Fachgebiet Intellektik, FachbereichInformatik, Technische Hochschule (Universität) Darmstadt, 1990

- The Relaxation of Unsolvable CSPs --- General Problem Formulationand Specific Illustration in the Scheduling Domain, IJCAI-89 Workshop onConstraint Processing, Workshop Proceedings (editor: Rina Dechter,Cognitive Systems Laboratory, UCLA, CA, U.S.A.), p. 154,11th International Joint Conference on Artificial Intelligence,Detroit, Michigan, U.S.A., August 1989

- Proper Constraint Relaxation, in João P. Martins andErnesto M. Morgado, editors, Proceedings (volume 2), EPIA-89,4th Portuguese Conference on Artificial Intelligence,Lisbon, Portugal, 1989

- Sensitive Relaxation of an Overspecified Constraint Network,Second International Symposium on Artificial Intelligence,Centro de Inteligencia Artificial, Instituto Tecnológico y deEstudios Superiores de Monterrey, Nuevo Léon, México, Proceedings,McGraw-Hill / Interamericana de México, 1989

- Parallel global constraint satisfaction,IJCAI-91 Workshop on Parallel Processing for Artificial Intelligence(PPAI-91), Informal Proceedings, pp. 80--85,12th International Joint Conference on Artificial Intelligence,Darling Harbour, Sydney, New South Wales, Australia,August 1991

- Massively distributed constraint satisfaction,Innovative Applications of Massive Parallelism,AAAI-1993 Spring Symposium, Stanford University, Palo Alto,Working Notes, pp. 99-105, Technical Report SS-93-04,AAAI Press, The American Association for Artificial Intelligence,Menlo Park, California, U.S.A., March 1993 (co-author)

- Notes on complexity issues within constraint satisfaction,IJCAI-93 Workshop on Knowledge-based Production Planning,Scheduling and Control, pp. 179–186,13th International Joint Conference on Artificial Intelligence,Chambéry, Savoie, France, August 1993 (co-author)

- Parallel distributed constraint satisfaction,IJCAI-93 Workshop on Parallel Processing for Artificial Intelligence(PPAI-93), 13th International Joint Conference on Artificial Intelligence,Chambéry, Savoie, France, August 1993 (co-author)

- A distributed realization for constraint satisfaction,in H. Kitano et al., editors, Volume 15 of Machine Intelligence andPattern Recognition series, Chapter 9, pp. 107–116, Elsevier Science,Amsterdam, The Netherlands, 1994 (co-author)

- Constraint Processing – Part I/II, Department of Mathematics andComputing Science, The University of the South Pacific, Suva, Fiji,March 1994

- On an improvement of a global algorithm for the NP-completeconstraint satisfaction problem, International Computer ScienceInstitute, Berkeley, California, USA, June 1996

- Research in Constraint-Based Layout, Visualization, CAD, and RelatedTopics: A Bibliographical Survey, DFKI Research Report RR-95-12,Deutsches Forschungszentrum für Künstliche Intelligenz, December 1995 +A bibliographical survey of constraint-based approaches to CAD,graphics, layout, visualization, and related topics, Knowledge-BasedSystems 9(7):449–464, November 1996, Elsevier Science (co-author)

- Reviewer of a project application for the EPSRC,Engineering and Physical Sciences Research Council, UK, 1996

- Fine-grained conflict resolution in constraint satisfaction problems,Journal of Experimental & Theoretical Artificial Intelligence,10(1):37–47, Taylor & Francis, January-March 1998,https://doi.org/10.1080/095281398146897, November 2010

- Global constraint satisfaction revisited,Technical Report TR-97-02, Department of Computer Science,University College Cork, National University of Ireland, 1997 +Revisiting global constraint satisfaction, Information ProcessingLetters 66(1):41-48, Elsevier Science, April 1998

- Computational Complexity in Constraint-based Combinatorial Auctions,INFORMATIK 2008, 38. Jahrestagung der Gesellschaft für Informatik,München, September 2008 (co-author)

- Boolean Algebra and Probability Theory, EU Erasmus guest professor,Angers, France, November 2008

His research interests are combinatorial optimization as well as cooperative and non-cooperative game theory.

A lively insight into presentations of the author (on the identical cardinality of N with just the natural even numbers as well as on the funny Monty Hall problem) could be possible – although in German – via https://www.youtube.com/watch?v=zeCCMOHVS3w.