Mathematical Foundations of Fuzzy Logic

• 26 hrs exercises
• 26 hrs projects

• 30 pts numeric exercises
• 70 pts projects

To extend an area of mathematical knowledge with an emphasis on solution searchings and mathematical problems proofs.
Successful students will gain deep knowledge of the selected area of mathematics (depending on the seminar group), and the ability to present the studied area and solve problems within it. The ability to understand advanced mathematical texts, the ability to design nontrivial mathematical proofs.

Classical logic only describes well the black and white world. Its consistent use in practical situations can lead to problems. This can be solved with multi-valued, e.g. fuzzy, logic which is the intuitive basis of any conjecture associated with vague terms. The modelling of fuzzy logic connectives is related to the study of the real variable functions. The mathematical apparatus required for the modelling of fuzzy logic connectives is the content of this course.

• Discrete Mathematics (IDM)
• Mathematical Analysis 1 (IMA1)

Knowledge of "IDA - Discrete Mathematics" and "IMA - Mathematical Analysis" courses.

• Carlsson, Ch., Fullér, R., Fuzzy reasoning in decision making and optimization, Studies in Fuzziness and Soft Computing, Vol. 82, 2002.

1. From classical logic to fuzzy logic
2. Modelling of vague concepts via fuzzy sets
3. Basic operations on fuzzy sets
4. Principle of extensionality
5. Triangular norms, basic notions, algebraic properties
6. Triangular norms, constructions, generators
7. Triangular conorms, basic notions and properties
8. Negation in fuzzy logic
9. Implications in fuzzy logic
10. Aggregation operators, basic properties
11. Aggregation operators, applications
12. Fuzzy relations
13. Fuzzy preference structures

1. Triangular norms, class of třída archimedean t-norms
2. Triangular norms, construction of continuous t-norms
3. Triangular norms, construction of non-continuous t-norms
4. Triangular conorms
5. Fuzzy negations and their properties
6. Implications in fuzzy logic
7. Aggregation operators, averaging operators
8. Aggregation operators, applications
9. Fuzzy relations, similarity, fuzzy equality
10. Fuzzy preference structures

• Active participation in the exercises (group problem solving, evaluation of the ten exercises): 30 points.
• Projects: group  presentation, 70 points.

• Active participation in the exercises (group problem solving, evaluation of the ten exercises): 30 points.
• Projects: group  presentation, 70 points.

Students have to get at least 50 points during the semester.