1. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20031 III. FUZZY LOGIC – Lecture 3 OBJECTIVES 1. To define the basic notions of fuzzy logic 2. To introduce the logical operations and relations on fuzzy sets 3. To learn how to obtain results of fuzzy logical operations 4. To apply what we learn to GIS

2. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20032 OUTLINE III. FUZZY LOGIC A. Introduction B. Inputs to fuzzy logic systems - fuzzification C. Fuzzy propositions D. Fuzzy hedges E. Computing the results of a fuzzy proposition given an input F. The resulting action

3. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20033 A. Introduction (figure from Earl Cox)

4. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20034 Introduction Steps (Earl Cox based on previous slide): 1. Input – vocabulary, fuzzification (creating fuzzy sets) 2. Fuzzy propositions – IF X is Y THEN Z (or Z is A) … there are four types of propositions 3. Hedges – very, extremely, somewhat, more, less 4. Combination and evaluation – computation of the results given the inputs 5. Action - defuzzification

5. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20035 Input – vocabulary, fuzzification (creating a fuzzy set) by using our previous methods of frequency, combination, experts/surveys (figure from Earl Cox)

6. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20036 Input (figure from Klir&Yuan)

7. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20037 Fuzzy Propositions – types 1 and 2 GENERAL FORMS 1. Unconditional and unqualified proposition: Q is P Example: Temperature(Q) is high(P) 2. Unconditional and qualified proposition: proposition(Q is P) is R Example: That Coimbra and Catania are beautiful is very true.

8. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20038 Fuzzy Proposition – type 1 and 2 (from Earl Cox)

9. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 20039 Fuzzy Propositions – type 1 and 2 (from Earl Cox)

10. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200310 Fuzzy Propositions – type 3 3. Conditional and unqualified proposition: IF Q is P THEN R is S Example: If Robert is tall, then clothes are large. If car is slow, then gear is low.

11. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200311 Fuzzy Propositions – type 4 4. Conditional and qualified proposition: IF Q is P THEN R is S is T {proposition(IF Q is P THEN R is S )} is T

12. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200312 Fuzzy Hedges (from Earl Cox)

13. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200313 Fuzzy Hedges (from Earl Cox)

14. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200314 Illustrations of Fuzzy Propositions – Composition/Evaluation (from Klir&Yuan)

15. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200315 Illustrations of Fuzzy Propositions – Composition/Evaluation (Earl Cox)

16. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200316 Illustrations of Fuzzy Propositions – Composition/Evaluation (from Earl Cox)

17. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200317 Illustrations of Fuzzy Propositions Decomposition – Defuzzification/Action (from Earl Cox)

18. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200318 Defuzzification (from Earl Cox)

19. August 12, 2003 III. FUZZY LOGIC: Math Clinic Fall 200319 Defuzzification (from Earl Cox)