Course detail

Fuzzy Sets and Applications

FSI-SFMAcad. year: 2019/2020

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.

Learning outcomes of the course unit

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.


Fundamentals of the set theory and mathematical analysis.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Klir, G. J. - Yuan, B.: Fuzzy Sets and Fuzzy Logic - Theory and Applications. New Jersey : Prentice Hall, 1995. (EN)
Novák, V.: Základy fuzzy modelování. Praha : BEN - technická literatura, 2000. (CS)
Kolesárová, A. - Kováčová, M.: Fuzzy množiny a ich aplikácie. Bratislava : STU, 2004. (CS)
Zimmermann, H. J.: Fuzzy Sets Theory and Its Applications. Boston : Kluwer-Nijhoff Publishing, 1998. (EN)
Talašová, J.: Fuzzy metody ve vícekriteriálním rozhodování a rozhodování. Olomouc : Univerzita Palackého, 2002. (CS)

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 topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

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).

Language of instruction


Work placements

Not applicable.


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.

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

Attendance at seminars is controlled and the teacher decides on the compensation for absences.

Classification of course in study plans

  • Programme M2A-P Master's

    branch M-MAI , 2. year of study, winter semester, 4 credits, compulsory

Type of course unit



26 hours, optionally

Teacher / Lecturer


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

13 hours, compulsory

Teacher / Lecturer


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.