Methods of Discrete Mathematics (FSI-SDM)

Academic year 2018/2019
Supervisor: prof. RNDr. Josef Šlapal, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
The course aims to acquaint the students with some usual methods of discrete mathematics used in various applications, especially in computer science.
Learning outcomes and competences:
The students will learn about the fundamentals of applied algebra. This will privide them with basic knowledge of the theory of ordered sets and lattices with an emphasis on Boolean algebras, the algebraic theory of automata and formal languages, and the coding theory.
Prerequisites:
Only the basic knowledge of the set theory is supposed that students acquired in high schools.
Course contents:
The subject Methods of discrete mathematics gets students acquainted with three basic areas of applied algebra. The first of them is the theory of ordered sets and lattices with the main stress focussed on the theory of Bolean algebras. The next area is the algebraic theory of automata and formal languages. The last one 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.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
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.
Controlled participation in lessons:
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.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Exercise  13 × 2 hrs. compulsory                  
Course curriculum:
    Lecture 1. Relations between sets
2. Mappings
3. Relations on a set
4. Tolerances and equivalences
5. Ordered sets
6. Lattices
7. Boolean lattices
8. Boolean functions
9. Applications of Boolean lattices
10.Formal languages
11.Finite automata
12.Grammars
13.Selfcorrecting codes
    Exercise 1. Relations between sets
2. Mappings
3. Relations on a set
4. Tolerances and equivalences
5. Ordered sets
6. Lattices
7. Boolean lattices
8. Boolean functions
9. Applications of Boolean lattices
10.Formal languages
11.Finite automata
12.Grammars
13.Selfcorrecting codes
Literature - fundamental:
1. N.L.Biggs, Discrete Mathematics, Oxford Univ. Press, 1999.
2. M.Piff, Discrete Mathematics, Cambridge Univ. Press, 1991.
3. A.D.Polimeni and H.J.Straight, Foundations of Discrete Mathematics, Brooks/Cole Publ. Comp., Pacific Grove, California, 1990.
4. D.R.Hankerson at al.: Coding Theory and Cryptography, Marcel Dekker, Inc., New York -Basel, 2000.
5. Steven Roman, Lattices and Ordered Sets, Springer, 2008.
Literature - recommended:
1. F. Preparata, R. Yeh: Úvod do teórie diskrétnych matematických štruktúr, Alfa, Bratislava, 1982.
2. M. Demlová, V. Koubek: Algebraická teorie automatů, SNTL, Praha, 1990.
3. J. Kopka: Svazy a Booleovy algebry, Univerzita J.E.Purkyně v Ústí nad Labem, 1991.
4. M.Novotný, S algebrou od jazyka ke gramatice a zpět, Academia, Praha, 1988.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
B3A-P full-time study B-MAI Mathematical Engineering -- Cr,Ex 5 Compulsory 1 2 W