Discrete Mathematics I (Discrete Algebraic Structures)

Discrete mathematics studies mathematical structures that are discrete rather than continuous. While the real numbers have the property of varying ''smoothly'', the objects studied in discrete mathematics do not vary in this way. Discrete means finite or countable (i.e., same cardinality as the set of natural numbers).

Research in discrete mathematics proliferated rather fast in the second half of the twentieth century due to the development of digital computers. Notations and concepts from discrete mathematics are particularly useful in studying and describing objects and problems in computer science. Conversely, discrete mathematics significantly benefitted from the invention of computers enabling the solution of difficult problems (e.g., four-coloring problem) or problem instances at a larger scale. Discrete Mathematics has a significant amount of applications in key areas like computer science, logistics, and life sciences.

The course will provide an introduction to discrete mathematics with an emphasis on discrete algebraic structures.

The class will take place

• Friday, 8:00 - 9:30 a.m., SBS 95, H0.16.

The class will start by October 29. By the way, the first-year students of Computational Informatics will have ''Orientierungswoche'' during this time. They will miss the first lecture on propositional calculus.

The labs will be held by Ralf Dittombee,

• Friday, 11:30 - 12:15 a.m., SBS 95, H0.07,

• Tuesday, 9:45 -  10:30 a.m., SBS 95, H0.09.

The labs will start by November 2. In each week, we will assign homework in the form of two or three problems that will be discussed in the labs. The assignments will be available in the form of PDF files via Stud.IP.

The organizer of the course will be Svetlana Torgasin. If you have any questions, please direct them to her.

The course will follow the book

• K.-H. Zimmermann: Diskrete Mathematik, BoD Verlag, Norderstedt, 2006.

There is an errata list and also a document with solutions of exercises. Both can be found at my homepage following the link on books. The TU library will eventually provide enough exemplars of the book.
Further books to be recommended:

• A. Beutelspacher, M. Zschiegner: Diskrete Mathematik für Einsteiger, Verlag Vieweg, 2007.

• N.L. Biggs: Discrete Mathematics, Oxford Univ. Press, 2002.

Contents:

• Introduction to higher mathematics: propositional calculus, first-order logic, sets, relations, equivalence and order relations, functions, natural numbers, countable sets.
• Combinatorics: combinations, permutations, counting principles.
• Algebraic structures: integers, rings, divisibility, residue class rings, polynomial rings, fields, finite fields.
• Applications: cryptography and coding theory.

The exam will be written. There will be a revision course directly before the exam.

Finally, I'd like to recommend some semi-mathematical books for recreation:

• D.R. Hofstadter: Gödel, Escher, Bach: ein Endloses Geflochtenes Band, Klett-Cotta, Stuttgart, 2008.

• S. Singh: Fermats letzter Satz, dtv Verlag, 2000.