Course detail
Fuzzy Sets and Applications
FSI-SFM-A Acad. year: 2025/2026 Winter semester
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
Supervisor
Department
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.
The study programmes with the given course
Programme N-MAI-A: Mathematical Engineering, Master's, compulsory
Type of course unit
Lecture
26 hours, optionally
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 hours, compulsory
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.