Course code Mate2010

Credit points 3

# Discrete Mathematics

Total Hours in Course81

Number of hours for lectures16

Number of hours for seminars and practical classes16

Independent study hours49

Date of course confirmation19.10.2022

Responsible UnitInstitute of Mathematics and Physics

prof.

## Natālija Sergejeva

Dr. math.

### Course abstract

The aim of the study course is to create an understanding of the basic concepts of discrete mathematics, which are necessary during the implementation of computational procedures, to teach how to operate with sets, to solve combinatorics problems, to simplify Boolean algebra expressions.
The study course provides the issues of modern discrete mathematics, including elements of set theory and mathematical logic, as well as some chapters of combinatorics (general counting methods for arrangements and selections, binomial coefficients, recurrence relations and elements of graph theory) and the axiom of mathematical induction. Study course contributes mathematical thinking, looks at various discrete mathematics applications in information technologies.

### Learning outcomes and their assessment

By the successful completion of this study course:
1. Students are able to manage and demonstrate knowledge and critical understanding of problems of set theory, mathematical logic and algebra of Boolean functions, general counting methods for arrangements and selections, mathematical induction, recurrence relations, elements of graph theory. Students manage the application of the acquired topics in practical examples related to their specialty. - tests.
2. Students are able to demonstrate understanding of relevant concepts and regularities, to perform necessary calculations and operations. – practical works.
3. Students are able to apply mathematical calculations corresponding to the specialty problem situation, to make a professional assessment and interpretation of the intermediate result of the calculations and the final results by working in the group or performing the work independently. - independent studies.

### Course Content(Calendar)

1. Basic notations of set theory. Operations with sets. Cartesian product of sets. - 2 h
2. Set algebra. Set identities. – 2 h
3. Mappings of functions, types of mappings. - 2 h
4. Relations, types of relations. – 2 h
5. Test: Set’s theory.
6. Propositions. Operations with propositions.
7. Boolean functions, duality of Boolean functions. – 2 h
8. Algebra of Boolean functions. Perfect normal forms. – 2 h
9. Representation of the Boolean functions with polynomials. Complete systems of Boolean functions. – 2 h
10. Simplification and minimization of Boolean functions. – 2 h
11. Boolean functions using for logic gates.
12. Predicates and their truth sets.
13. Test: Mathematical logic.
14. Mathematical induction. - 2 h
15. Combinatorics. Pascal’s triangle and binomial coefficients. – 2 h
16. Recurrence relations. Solving of relations. – 2 h
17. Basic notations of graph theory. Graphs and matrices. - 2 h
18. Graph isomorphism problem. Cycles and trees. Regular graphs. Graph Planarity. - 2 h
19. Test: Combinatorics, mathematical induction and elements of graph theory.

### Description of the organization and tasks of students’ independent work

The following independent works must be completed in writing form:
Independent work 1: Elements of set theory
Independent work 2: Mathematical logic
Independent work 3: Mathematical Induction,
combinatorics

### Criteria for Evaluating Learning Outcomes

The course is completed without additional knowledge examination if the results of the semester are summarized as all independent works are completed (all tasks are completed correctly); during the semester each test score is at least 4.
Failed tests can be repeated during the study process at the time indicated by the academic staff. The student can repeat the last test in the 1st week of period of individual studies and examinations at the time indicated by the academic staff.
In the case of unsuccessful work in the semester student answers for all the topics together in the period of the individual studies and examinations at the time indicated by the academic staff.

### Compulsory reading

1. Daugulis P. Diskrētā matemātika. Rēzekne: Rēzeknes Augstskolas Izdevniecība, 2001.
2. Diskrētā matemātika uzdevumos un piemēros. Rīgas Tehniskā universitāte. Inženiermatemātikas katedra; [sast. I. Volodko]. Rīga: RTU izd., 2004. - 126 lpp.
3. Strazdiņš I. Diskrētā matemātika. Rīga: Zvaigzne ABC, 2001. - 148 lpp.
4. Erciyes K. Discrete Mathematics and Graph Theory: A Concise Study Companion and Guide. Springer Nature Switzerland AG, 2021. - 336 p. (pieejama Matemātikas katedrā/ available in the Department of Mathematics)

### Further reading

1. Garnier R. Discrete mathematics for new technology. Bristol: Philadelphia, Institute of Physics Publishing, 1999. – 678 p.
2. Volodko I. Tipveida uzdevumu krājums diskrētajā matemātikā. R: RTU: 2002. - 62 lpp.

### Periodicals and other sources

Kanders U., Andžāns A. Matemātiskās indukcijas tālmācības kurss. http://www.lanet.lv/info/matind/ [tiešsaiste]. [skatīts 05.12.2018.]

### Notes

Compulsory course for Faculty of Information Technologies Bachelor’s study programmes “Computer Control and Computer Science” and „Information Technologies for Sustainable Development”.