Course detail

Fuzzy Sets and Applications

FSI-SFM-AAcad. year: 2025/2026

The course is concerned with the fundamentals of the fuzzy sets theory: operations with fuzzy sets, extension principle, fuzzy numbers, fuzzy relations and graphs, fuzzy functions, linguistics variable, fuzzy logic, approximate reasoning and decision making, fuzzy control, fuzzy probability. It also deals with the applicability of those methods for modelling of vague technical variables and processes, and work with special software of this area.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

doc. RNDr. Libor Žák, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Fundamentals of the set theory and mathematical analysis.

Rules for evaluation and completion of the course

Course-unit credit requirements: active participation in seminars, mastering the subject matter, passing all tests. Examination (written form) consists of two parts: a practical part (4 tasks related to: operations with fuzzy sets, unary and binary operations with fuzzy numbers, fuzzy relation, fuzzy function, fuzzy logic, fuzzy control) using the summary of formula; theoretical part (4 tasks related to basic notions, their properties, sense and practical use); evaluation: each task 0 to 20 points and each theoretical question 0 to 5 points; evaluation according to the total number of points (scoring 0 points for any theoretical part task means failing the exam): excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).
Attendance at seminars is controlled and the teacher decides on the compensation for absences.

Aims

The course objective is to make students acquainted with basic methods and applications of fuzzy sets theory, that allows to model vague quantity of numerical and linguistic character, and subsequently systems and processes, which cannot be described with classical mathematical models. A part of the course is the work with fuzzy toolbox of software Matlab and shareware products.
Students acquire necessary knowledge of important parts of fuzzy set theory, which will enable them to create effective mathematical models of technical phenomena and processes with uncertain information, and carry them out on PC by means of adequate implementations.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

GOTTWALD, S.: Fuzzy Set and Fuzzy Logic. Verlag Vieweg, Wiesbaden 1993. (EN)
Klir, G. J. - Yuan, B.: Fuzzy Sets and Fuzzy Logic - Theory and Applications. New Jersey : Prentice Hall, 1995. (EN)
Novák, V.: Fuzzy množiny a jejich aplikace. Praha : SNTL, 1990. (CS)
Zimmermann, H. J.: Fuzzy Sets Theory and Its Applications. Boston : Kluwer-Nijhoff Publishing, 1998. (EN)

Recommended reading

Klir, George Jiří; Folger, Tina A. (1988). Fuzzy sets, uncertainty, and information. Englewood Cliffs, NJ: Prentice Hall. ISBN 978-0-13-345984-5. (EN)
Kolesárová, A. - Kováčová, M.: Fuzzy množiny a ich aplikácie. Bratislava : STU, 2004. (CS)
Novák, V.: Základy fuzzy modelování. Praha : BEN - technická literatura, 2000. (CS)
Talašová, J.: Fuzzy metody ve vícekriteriálním rozhodování a rozhodování. Olomouc : Univerzita Palackého, 2002. (CS)

Classification of course in study plans

  • Programme N-MAI-A Master's 2 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. RNDr. Libor Žák, Ph.D.

Syllabus

1. Fuzzy sets (motivation, basic notions, properties).
2. Operations with fuzzy sets (properties).
3. Operations with fuzzy sets (alfa cuts).
4. Triangular norms and co-norms, complements (properties).
5. Extension principle (Cartesian product, extension mapping).
6. Fuzzy numbers (definition, extension operations, interval arithmetic).7. Fuzzy relations (basic notions, kinds).
8. Fuzzy functions (basic orders, fuzzy parameter, derivation, integral).
9. Linguistic variable (model, fuzzification, defuzzification).
10. Fuzzy logic (multiple value logic, extension).
11. Approximate reasoning and decision-making (fuzzy environment, fuzzy control).
12. Fuzzy probability (basic notions, properties).
13. Fuzzy models design for applications.

Computer-assisted exercise

26 hod., compulsory

Teacher / Lecturer

doc. RNDr. Libor Žák, Ph.D.

Syllabus

1. Sets, relations and operations.
2. Fuzzy sets (basic notions, properties).
3. Operations with fuzzy sets (properties, alfa cuts).
4. Triangular norms and co-norms, complements.
5. Extension principle of mapping.
6. Fuzzy numbers (extension unary and binary operations).
7. Fuzzy numbers and interval arithmetic.
8. Fuzzy relations (orders, operations).
9. Fuzzy functions with fuzzy parameter (derivation, integral).
10. Linguistic variable (operators, presentation).
11. Fuzzy logic (operations, properties).
12. Approximate reasoning and decision-making (fuzzy control).
13. Applications of fuzzy models.