Title:

# Mathematical Foundations of Fuzzy Logic

Code:IMF
Ac.Year:2018/2019
Sem:Winter
Curriculums:
ProgrammeField/
Specialization
YearDuty
BIT-2ndElective
IT-BC-3BIT-Elective
IT-BC-3BIT2ndElective
Language of Instruction:Czech
Credits:5
Completion:classified credit
Type of
instruction:
Hour/semLecturesSeminar
Exercises
Laboratory
Exercises
Computer
Exercises
Other
Hours:0260026
ExamsTestsExercisesLaboratoriesOther
Points:0030070
Guarantor:Hliněná Dana, doc. RNDr., Ph.D. (DMAT)
Deputy guarantor:Havlena Vojtěch, Ing. (DITS)
Instructor:Hliněná Dana, doc. RNDr., Ph.D. (DMAT)
Faculty:Faculty of Electrical Engineering and Communication BUT
Department:Department of Mathematics FEECS BUT
Prerequisites:
 Discrete Mathematics (IDA), DMAT Mathematical Analysis (IMA), DMAT

Learning objectives:
To extend an area of mathematical knowledge with an emphasis of solution searchings and mathematical problems proofs.
Description:
At the beginning of semester, students choose from the supplied topics. On the weekly seminars, they present the topics and discuss about them. The final seminar is for assesment of students' performance.
Knowledge and skills required for the course:
Knowledge of "IDA - Discrete Mathematics" and "IMA - Mathematical Analysis" courses.
Subject specific learning outcomes and competencies:
Successfull students will gain deep knowledge of the selected area of mathematics (depending on the seminar group), and ability to present the studied area and solve problems within it.
Generic learning outcomes and competencies:
The ability to understand advanced mathematical texts, the ability to design nontrivial mathematical 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 modeling of fuzzy logic connectives is related to the study of the real variable functions. The mathematical apparatus required for the modeling of fuzzy logic connectives is the content of this course.
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
Fundamental literature:

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

1. Alsina, C., Frank, M.J., Schweizer, B., Assocative functions: Triangular Norms and Copulas, World Scientific Publishing Company, 2006
2. Kolesárová, A., Kováčová, M., Fuzzy množiny a ich aplikácie, STU v Bratislave, 2004
3. Trillas, E., Eciolaza, L, Fuzzy logic-An introductory course for engineering students, Studies in Fuzziness and Soft Computing, 2015
Controlled instruction:
Active participation in the exercises (group problem solving, evaluation of the ten exercises): 30 points.
Projects: group  presentation, 70 points.
Progress assessment:
Active participation in the exercises: 30 points.
Projects: 70 points.
Exam prerequisites:
Students have to get at least 50 points during the semester.