chapter7 - Chapter 7 Fuzzy Sets and Operations Fuzziness...

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Chapter 7 Fuzzy Sets and Operations
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Chapter 7 --- Fuzzy Sets and Operations 2 Fuzziness versus Probability c Fuzziness : an alternative to randomness for describing uncertainty. c Fuzzy Theory : all things admit degrees (not clear cut), but admit them deterministically. c Fuzziness & Randomness differ conceptually & theoretically.
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Chapter 7 --- Fuzzy Sets and Operations 3 Fuzziness versus Probability c Similar points between fuzziness & randomness: c Both describe uncertainty with numbers in the unit interval [0,1]. c Both systems combine sets and propositions associatively, commutatively and distributively. c Main difference : how the system jointly treat a set A and its opposite A C . c Probability: P ( A A C ) = P ( φ ) = 0 c Fuzziness: A A C 0. φ : Empty Set
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Chapter 7 --- Fuzzy Sets and Operations 4 Randomness vs. Ambiguity : whether vs. how much c Randomness: describes the uncertainty of event occurrence whether there will be rain tomorrow. c Fuzziness: describes event ambiguity, it measures the degree to which an event occurs, not whether it occurs. It rains. But is the rain heavy? Example : Example :
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Chapter 7 --- Fuzzy Sets and Operations 5 Randomness vs. Ambiguity : whether vs. how much c Fuzziness arises from the ambiguity or vagueness between a thing A and its opposite A C . c Non-degenerate overlap: A A C 0 breaks the “Law of Non-Contradiction” c Non-degenerate underlap: A A C X breaks the “Law of Excluded Middle” X : The whole set
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Chapter 7 --- Fuzzy Sets and Operations 6 Example 1 Set A = Men, Set A C = Women c Law of Non-Contradiction : nobody can be both a man and a woman, i.e., A A C = 0. c Law of Excluded Middle : all men and women form the whole set of human beings, i.e., A A C = X . No fuzzy sense because there’s no ambiguity between set A and its opposite A C . The definition is clear .
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Chapter 7 --- Fuzzy Sets and Operations 7 Example 2 Set A = Beautiful, Set A C = Ugly c A person can be considered as beautiful (to some extent) by someone but ugly (to some extent) by others. Moreover, sometimes he/she is beautiful but sometimes ugly. A A C 0. c Besides beautiful and ugly people, there are some very typical and ordinary ones (neither beautiful nor ugly) A A C X . have fuzzy sense because there’s ambiguity between set A and its opposite A C . It is not clear cut .
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Chapter 7 --- Fuzzy Sets and Operations 8 Geometry of Fuzzy Sets: Sets as Points c The fuzzy sense can be represented in geometry, by a square (2-D) or a n -D hybercube I n = [0,1] n . c
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chapter7 - Chapter 7 Fuzzy Sets and Operations Fuzziness...

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