Course detail

Methods of Discrete Mathematics

FSI-SDMAcad. year: 2022/2023

The subject Methods of discrete mathematics gets students acquainted with basic areas of set theory, discrete mathematics, and applied algebra. The first area is formed by relations between sets and on sets with a stress on partially ordered sets. The next area covers Axiom of Choice and cardinal and ordinal numbers. After that, lattice theory is discussed with the main interest focused on the theory of Bolean algebras. Then the algebraic theory of automata and formal languages follows. The last area is an introduction to the coding theory. Thus, all the three areas represent theoretical fundamentals of informatics. With respect to the expansion of using computers in all branches of engineering, the acquired knowledge is necessary for graduates in mathematical engineering.

Language of instruction

Czech

Number of ECTS credits

6

Mode of study

Not applicable.

Guarantor

prof. RNDr. Josef Šlapal, CSc.

Department

Institute of Mathematics (ÚM)

Learning outcomes of the course unit

The students will learn about the fundamentals of ordered sets, lattices and Boolean algebras. They will learn to minimize Boolean functions and realize them by logic circuits. They will also get acquainted with the most usual types of automata and their properties, with regular languages and with the problem of determinism. Finally, thay will also get an image of the basic problems connected with coding and decoding of messages..

Prerequisites

Only the basic knowledge of the set theory is supposed that students acquired in high schools.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical getting through with topics presented in lectures by solving problems and exercises.

Assesment methods and criteria linked to learning outcomes

The course unit credit is awarded on condition of having attended the seminars actively and passed a written test. The exam has a written and an oral part. The written part tests student's ability to deal with various problems using the knowledge and skills acquired in the course. In the oral part, the student has ro prove that he or she has mastered the related theory.

Course curriculum

Not applicable.

Work placements

Not applicable.

Aims

The course aims to acquaint the students with some usual methods of discrete mathematics used in various applications, especially in computer science for construction and process description of computers and for data transmission. The students will get a proof that, beside the continuous mathematics, also discrete mathematics is a basic discipline whose knowledge is a necessary condition for a successful creative work of an engineer.

Specification of controlled education, way of implementation and compensation for absences

The attendance at seminars is required and will be checked regularly by the teacher supervising a seminar. If a student misses a seminar due to excused absence, he or she will receive problems to work on at home and catch up with the lessons missed.

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

A.D.Polimeni and H.J.Straight, Foundations of Discrete Mathematics, Brooks/Cole Publ. Comp., Pacific Grove, California, 1990. (EN)
D.R.Hankerson at al.: Coding Theory and Cryptography, Marcel Dekker, Inc., New York -Basel, 2000. (EN)
M.Piff, Discrete Mathematics, Cambridge Univ. Press, 1991. (EN)
N.L.Biggs, Discrete Mathematics, Oxford Univ. Press, 1999. (EN)
Steven Roman, Lattices and Ordered Sets, Springer, 2008. (EN)

Recommended reading

F. Preparata, R. Yeh: Úvod do teórie diskrétnych matematických štruktúr, Alfa, Bratislava, 1982.
J. Kopka: Svazy a Booleovy algebry, Univerzita J.E.Purkyně v Ústí nad Labem, 1991.
M. Demlová, V. Koubek: Algebraická teorie automatů, SNTL, Praha, 1990.
M.Novotný, S algebrou od jazyka ke gramatice a zpět, Academia, Praha, 1988.

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 2 year of study, winter semester, compulsory

  • Programme LLE Lifelong learning

    branch CZV , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Relations between sets and on sets
2. Tolerances, equivalences, preorders and orders
3. Ordered sets
4. Axiom of choice and equivalent statements
5. Ordinal and cardinal numbers
6. Lattices, irreducibility, ideals and filters
7. Boolean lattices and functions, applications
8. Complete lattices, closure operators
9. Galois connections, Dedekind-MacNeille completion
10.Formal languages
11.Finite automata
12.Grammars
13.Selfcorrecting codes

Exercise

26 hod., compulsory

Teacher / Lecturer

prof. RNDr. Josef Šlapal, CSc.

Syllabus

1. Relations between sets and on sets
2. Tolerances, equivalences, preorders and orders
3. Ordered sets
4. Axiom of choice and equivalent statements
5. Ordinal and cardinal numbers
6. Lattices, irreducibility, ideals and filters
7. Boolean lattices and functions, applications
8. Complete lattices, closure operators
9. Galois connections, Dedekind-MacNeille completion
10.Formal languages
11.Finite automata
12.Grammars
13.Selfcorrecting codes