Course details

# Discrete Mathematics

IDM Acad. year 2020/2021 Winter semester 5 credits

Guarantor

Deputy Guarantor

Language of instruction

Completion

Time span

Assessment points

Department

Lecturer

Hliněná Dana, doc. RNDr., Ph.D. (DMAT FEEC BUT)

Holík Lukáš, Mgr., Ph.D. (DITS FIT BUT)

Lengál Ondřej, Ing., Ph.D. (DITS FIT BUT)

Instructor

Fuchs Petr, RNDr., Ph.D. (DMAT FEEC BUT)

Havlena Vojtěch, Ing. (DITS FIT BUT)

Hliněná Dana, doc. RNDr., Ph.D. (DMAT FEEC BUT)

Holík Lukáš, Mgr., Ph.D. (DITS FIT BUT)

Lengál Ondřej, Ing., Ph.D. (DITS FIT BUT)

Svoboda Zdeněk, RNDr., CSc. (DMAT FEEC BUT)

Vážanová Gabriela, Mgr. (DMAT FEEC BUT)

Subject specific learning outcomes and competences

Learning objectives

Why is the course taught

Discrete mathematics aims at understanding the aspects of the real world that are the most fundamental from the point of view of computer science. It studies such concepts as a set (e.g. a collection of data, resources, agents), relations and graphs (e.g. relationships among data or description of a communication), and operations over elements of a set (especially basic arithmetical operations and their generalization).

The mathematical logic then gives means of expressing ideas and reasoning clearly and correctly and is, moreover, the foundation of "thinking of computers".

Generally speaking, discrete mathematics teaches the art of abstraction -- how to apprehend the important aspects of a problem and work with them. It provides a common language for talking about those aspects precisely and effectively. Besides communication of ideas, it helps to structure thought into exactly defined notions and relationships, which is necessary when designing systems so large and complex as today's software and hardware.

For example, discrete math gives the basic tools for expressing what a program does; what its data structures represent; how the amount of needed resources depends on the size of the input; how to specify and argue that a program does what it should do. Similarly essential uses can be found everywhere in computer science. One could say that a programmer without mathematics is similar to a piano player who cannot read notes: if he is talented, he can still succeed, but his options are limited, especially when it comes to solving complex problems.

In order to teach mathematical thinking to students, we emphasize practising mathematics by using it to solve problems -- in the same way as programming can be only learnt through programming, mathematics also can be learnt only by doing it.

Prerequisite kwnowledge and skills

Study literature

- Matoušek J., Nešetřil J.,
*Kapitoly z diskrétní matematiky*, Karolinum, Praha 2007. (in Czech). - Matoušek J., Nešetřil J.,
*Invitation to Discrete Mathematics*, Oxford University Press, Oxford 2008. - Hliněný, P., Úvod do informatiky.
*Elportál*, Brno, 2010. (in Czech). - Kovár, M., Diskrétní matematika, FEKT VUT, Brno, 2013. (in Czech).

Fundamental literature

- Anderson I.,
*A First Course in Discrete Mathematics*, Springer-Verlag, London 2001. - Grimaldi R. P.,
*Discrete and Combinatorial Mathematics*, Pearson Addison Valley, Boston 2004. - Grossman P.,
*Discrete mathematics for computing*, Palgrave Macmillan, New York 2002. - Kolibiar, M. a kol.,
*Algebra a príbuzné disciplíny*, Alfa, Bratislava, 1992. (in Slovak). - Kolman B., Busby R. C., Ross S. C.,
*Discrete Mathematical Structures*, Pearson Education, Hong-Kong 2001. - Klazar M., Kratochvíl J, Loebl M., Matoušek J. Thomas R., Valtr P.,
*Topics in Discrete Mathematics*, Springer-Verlag, Berlin 2006. - Matoušek J., Nešetřil J.,
*Kapitoly z diskrétní matematiky*, Karolinum, Praha 2007. (in Czech). - Matoušek J., Nešetřil J.,
*Invitation to Discrete Mathematics*, Oxford University Press, Oxford 2008. - O'Donnell, J., Hall C., Page R.,
*Discrete Mathematics Using a Computer*, Springer-Verlag, London 2006. - Sochor, A.,
*Klasická matematická logika*, Karolinum, Praha 2001. (in Czech).

Syllabus of lectures

- The formal language of mathematics. A set intuitively. Basic set operations. Power set. Cardinality. Sets of numbers. The principle of inclusion and exclusion.
- Binary relations and mappings. The composition of binary relations and mappings.
- Reflective, symmetric, and transitive closure. Equivalences and partitions.
- Partially ordered sets and lattices. Hasse diagrams. Mappings.
- Binary operations and their properties.
- General algebras and algebras with one operation. Groups as algebras with one operation. Congruences and morphisms.
- General algebras and algebras with two operations. Lattices as algebras with two operations. Boolean algebras.
- Propositional logic. Syntax and Semantics. Satisfiability and validity. Logical equivalence and logical consequence. Ekvivalent formulae. Normal forms.
- Predicate logic. The language of first-order predicate logic. Syntax, terms, and formulae, free and bound variables. Interpretation.
- Predicate logic. Semantics, truth definition. Logical validity, logical consequence. Theories. Equivalent formulae. Normal forms.
- A formal system of logic. Hilbert-style axiomatic system for propositional and predicate logic. Provability, decidability, completeness, incompleteness.
- Basic concepts of graph theory. Graph Isomorphism. Trees and their properties. Trails, tours, and Eulerian graphs.
- Finding the shortest path. Dijkstra's algorithm. Minimum spanning tree problem. Kruskal's and Jarnik's algorithms. Planar graphs.

Syllabus of numerical exercises

Progress assessment

- Evaluation of the two home assignments solved in the groups (max 10 points).
- Evaluation of the two mid-term exams (max 30 points).

Controlled instruction

- The knowledge of students is tested at exercises; including two homework assignments worth for 5 points each, at two midterm exams for 15 points each, and at the final exam for 60 points.
- If a student can substantiate serious reasons for an absence from an exercise, (s)he can either attend the exercise with a different group (please inform the teacher about that).
- Passing boundary for ECTS assessment: 50 points.

Exam prerequisites

Course inclusion in study plans