Course details

# Mathematical Foundations of Fuzzy Logic

IMF Acad. year 2022/2023 Winter semester 5 credits

At the beginning of the semester, students choose from the supplied topics. On the weekly seminars, they present the topics and discuss them. The final seminar is for assessment of students' performance.

Guarantor

Course coordinator

Language of instruction

Czech, English

Completion

Classified Credit

Time span

• 26 hrs exercises
• 26 hrs projects

Assessment points

• 30 pts numeric exercises
• 70 pts projects

Department

Instructor

Subject specific learning outcomes and competences

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.

Learning objectives

To extend an area of mathematical knowledge with an emphasis on solution searchings and mathematical problems proofs.

Why is the course taught

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.

Prerequisites

Prerequisite knowledge and skills

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

Study literature

• Alsina, C., Frank, M.J., Schweizer, B., Assocative functions: Triangular Norms and Copulas, World Scientific Publishing Company, 2006.
• Baczynski, M., Jayaram, B., Fuzzy implications, Studies in Fuzziness and Soft Computing, Vol. 231, 2008.
• Carlsson, Ch., Fullér, R., Fuzzy reasoning in decision making and optimization, Studies in Fuzziness and Soft Computing, Vol. 82, 2002.
• Carlsson, Ch., Fullér, R., Fuzzy reasoning in decision making and optimization, Studies in Fuzziness and Soft Computing, Vol. 82, 2002.
• Kolesárová, A., Kováčová, M., Fuzzy množiny a ich aplikácie, STU v Bratislave, 2004.
• Trillas, E., Eciolaza, L, Fuzzy logic-An introductory course for engineering students, Studies in Fuzziness and Soft Computing, 2015.

Syllabus of numerical exercises

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

Syllabus - others, projects and individual work of students

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

Progress assessment

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

Controlled instruction

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

Exam prerequisites

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

Schedule

DayTypeWeeksRoomStartEndCapacityLect.grpGroupsInfo
Thu exercise 1., 2., 3., 4., 5., 6., 7., 8., 10., 11., 12., 13. of lectures A113 08:0009:5064 2BIA 2BIB 3BIT xx Hliněná

Course inclusion in study plans

• Programme BIT, 2nd year of study, Elective
• Programme BIT (in English), 2nd year of study, Elective
• Programme IT-BC-3, field BIT, 2nd year of study, Elective