2. Fuzzy Aspects of Systems Science William Silvert IPIMAR, Portugal

3. Fuzzy and Crisp Sets A traditional (crisp) set In Out

4. Fuzzy and Crisp Sets A fuzzy set In Out ???

5. Does Fuzzy = Fantastic? Some people think that fuzzy set theory is absolutely great, Some dont. the best idea since the Arabs invented zero.

6. Literature on Fuzzy Sets zAlles das ist nicht Politik ist Nichtpolitik. Thomas Mann zYou have seen him spout; then declare what the spout is; can you not tell water from air? My dear sir, in this world it is not so easy to settle these plain things. Herman Melville

7. And Current Thinking? Liquid–vapour interfaces, particularly those involving water, are common in both natural and artificial environments. They were first described as regions of continuous variation of density, caused by density fluctuations within the bulk phases. Fradin et al. In Nature, 24 Feb. 2000

8. Legality of Fuzzy Sets

9. Even Sex is Fuzzy zCan an organism go from female to male in an instant? No, at some point it is part female and part male. zWe usually say that an organism is male or female, but this doesnt work for those (like corn) that possess both male and female organs, or that change sex.

10. Remote Sensing Land Water (Marsh)

11. What is Land? 100% 0% 40%

12. Ideal Food Chain Plants Herbivores Carnivores TL = 0 TL = 1 TL = 2

13. Realistic Food Web Plants Herbivores Omnivores TL = 0 TL = 1 TL = ? +

14. The Ecological Niche 10 o Temperature 20 o

15. Factory Emissions Threshold Level 95% 55% 45%

16. Ambiguous Data

17. Temperature Control zWe can design a thermal controller that adjusts the heater output by a formula such as Heat = F (Temperature, Rate of Change) zWe can also design a fuzzy controller that does not use a continuous function to control the heating.

18. Very Fuzzy Control

19. Typical Fuzzy Control

20. Forecasting zA forecast requires specification of a set. zIf we predict that a pair of dice will come up with ten, then of the 36 (6 x 6) possibilities, 3 are in the prediction set - {6+4}, {5+5}, {4+6}

21. Fuzzy Forecasting zMost forecasts do not use crisp sets. zIf we say that the dice are loaded and favour high numbers, how can we check that? (For this example we can ignore the obvious statistical test).

22. A Fuzzy Prediction Set The prediction set of high values for a throw of two dice

23. Are Fuzzy Sets Unusual? zAlles das ist nicht Fuzzy ist Nichts. zMost categories used in science (and in everything else) are fuzzy, whether we say so or not. zEven astronomy is fuzzy: what about Red Giants? zand psychology - what about borderline schizophrenics?

24. Summary zFuzzy set theory is a natural way to explain most natural categories. zFuzzy set theory is not especially fancy from a mathematical point of view. zObjections to fuzzy thinking seem to be more psychological than scientific. zSo give it a try!