Discrete Structures

The lecture introduces elementary concepts and important areas of discrete mathematics that are relevant for informatics students. It covers the following five topics:

1) Basic concepts of sets, relations and functions:
- sets: basic operations, equivalence laws, KV-diagram, countable and uncountable sets, Cantor's Theorem
- relations: join, transitive hull, relational algebra
- functions: basic properties, composition, inverse

2) Fundamentals of Propositional Logic and First-Order Logic:
- Propositional Logic:
- syntax and semantics
- truth tables and their connection to KV-diagrams
- equivalence laws
- CNF, DNF, normalization procedure, equisatisfiability
- SAT-procedure: DPLL, resolution, proof of correctness
- modelling with propositional logic
- Predicate Logic:
- syntax and semantics
- equivalence laws
- modelling with predicate logic

3) Basics of combinatorics:
- counting principles
- drawing of balls from urns: variations, permutations, combinations
- binomial coefficients: symmetry, identities of Pascal and Vandermonde
- distribution problems
- Stirling-numbers of the first and second kind
- ordered and unordered partition functions
- application: load distribution

4) Basics of graph theory:
- basic definitions
- trees
- Euler and Hamilton circuits: Euler's theorem, theorems of Dirac and Ore
- planar graphs: Euler's polyhedron formula, Kuratowski's theorem
- matchings: marriage theorem, augmenting paths
- matchings with preferences: Gale-Shapley's theorem

5) Algebraic basics:
- basic definitions: algebra, group, ring, field
- groups:
- order: Lagrange's theorem, generator, group exponent
- cyclic groups
- basics of number theory: largest common divisor, extended euclidean
algorithm, Euler's phi function
- multiplicative groups of integers modulo n
- RSA

On successful completion of the module, students will be able to
- understand the elementary vocabulary of discrete mathematics and use logic, algebraic und algorithmic calculi,
- solve combinatoric problems,
- model and solve problems using graph theory, and
- do a quantitive analysis of the efficiency of algorithms.

none

The module consists of lectures and exercises. During the lectures students are asked to solve small exercises online. Students also receive weekly assignments, whose solution is discussed in the exercises.

Slide show, blackboard, written assignments.

- A. Steger: Diskrete Strukturen, Band 1: Kombinatorik, Graphentheorie, Algebra, Springer, 2001
- K.H. Rosen: Discrete Mathematics And Its Applications, McGraw-Hill, 1995
M. Aigner: Diskrete Mathematik, Vieweg, 1999 (3rd Edition)
- R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics: a Foundation for Computer Science, Addison-Wesley, 1994
- D. Gries, F.B. Schneider: A Logical Approach to Discrete Math, Springer, 1993
- D.L. Kreher, D.R. Stinson: Combinatorial Algorithms: Generation, Enumeration, and Search, CRC Press, 1999
- S. Pemmaraju, S. Skiena: Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica, Cambridge University Press, 2003
- Schöning, Uwe: Logik für Informatiker, Spektrum-Verlag, 2000 (5. Auflage)

Students are assessed by means of a written 120 - 180 minutes exam consisting of a list of exercises.
Some exercises test if the student can correctly use the mathematical vocabulary about sets, relations, logic, graphs and other mathematical objects introduced in the lectures. Further exercises test if the student is able to select the right logical, combinatorial, graph theoretical, or algebraic technique for the solution of a specific problem, and can apply it correctly.