Diskret matematik

**Valid from:** Autumn 2021**Decided by:** FN1/Anders Gustafsson**Date of establishment:** 2013-11-15

**Division:** Mathematics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FMA091**Teaching language:** English

The aim of the course is to treat some basic parts of discrete mathematics, of importance in computer science, information theory, signal processing, physics and many other subjects in technology and science. The aim is also to develop the students' ability to solve problems and to assimilate mathematical text. The course should also provide general mathematical education.

*Knowledge and Understanding*

For a passing grade the doctoral student must

- be able to understand and in his or her own words clearly define the central concepts in combinatorics, number theory, graph theory, functions and relations, and the theory of finite fields.
- in his or her own words be able to describe the logical connections between the occurring concepts (theorems and proofs).
- with confidence be able to carry out routine calculations within the framework of the course.
- in practical situations, with confidence be able to identify different combinatorial selections: with/without repetition, with/without regard to order.
- understand how results about finite fields and linear algebra may be used for coding.

*Competences and Skills*

For a passing grade the doctoral student must

- be able to demonstrate ability to identify problems which can be solved with methods from discrete mathematics and to choose an appropriate method.
- in connection with problem solving be able to demonstrate ability to integrate results from various parts of the course.
- with proper terminology, in a well-structured way and with clear logic be able to explain the solution to a problem.
- be able to use basic theorems of graph theory to draw conclusions about a given graph (of moderate size).

Number theory: Divisibility. Prime numbers. The Euclidean algorithm. Chinese remainder theorem. Modular arithmetic. Sets, functions and relations: Injective, surjective and bijective functions. Inverse function. Equivalence relations. Combinatorics: The four cases of counting with or without repetition and with or without regard to order. Binomial coefficients. The principle of inclusion and exclusion. The method of generating functions. Recursion: Recursion formulae and difference equations. Rings and fields: Definition. Applications to coding. Graph theory: Terminology and basic concepts. Eulerian and Hamiltonian graphs.

- Grimaldi, Ralph P.: Discrete and Combinatorial Mathematics: Pearson New International Edition. 2013. ISBN 9781292022796.
- Andersson, K.: Finite Fields and Error-Correcting Codes.. Matematikcentrum, 2015.

Available as a pdf-file on the web. 54 pages.

**Types of instruction:** Lectures, seminars, exercises

**Examination formats:** Written exam, oral exam**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Elementary linear algebra and analysis (FMAB65 and FMAB20).

**Course coordinators:** **Web page:** http://www.maths.lth.se/utbildning/matematiklth/