1. Fuzzy sets - Basic Definitions1 Classical (crisp) set A collection of elements or objects x  X which can be finite, countable, or overcountable. A collection of elements or objects x  X which can be finite, countable, or overcountable. A classical set can be described in two way: A classical set can be described in two way: Enumerating (list) the elements ； describing the set analytically Enumerating (list) the elements ； describing the set analytically Example: stating conditions for membership --- {x|x  5} Example: stating conditions for membership --- {x|x  5} Define the member elements by using the characteristic function, in which 1 indicates membership and 0 nonmembership. Define the member elements by using the characteristic function, in which 1 indicates membership and 0 nonmembership.

2. Fuzzy sets - Basic Definitions2 Fuzzy set If X is a collection of objects denoted generically by x then a fuzzy set in X is a set of ordered pairs: If X is a collection of objects denoted generically by x then a fuzzy set in X is a set of ordered pairs: is called the membership function or grade of membership of x in

3. Fuzzy sets - Basic Definitions3 Example 1 1 A realtor wants to classify the house he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in it. Let X={1,2,3,…,10} be the set of available types of houses described by x=number of bedrooms in a house. Then the fuzzy set “comfortable type of house for a 4- person family “may be described as A realtor wants to classify the house he offers to his clients. One indicator of comfort of these houses is the number of bedrooms in it. Let X={1,2,3,…,10} be the set of available types of houses described by x=number of bedrooms in a house. Then the fuzzy set “comfortable type of house for a 4- person family “may be described as

4. Fuzzy sets - Basic Definitions4 Example 1 2 ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

5. Fuzzy sets - Basic Definitions5 Example 2 =“real numbers considerably larger than 10” =“real numbers considerably larger than 10” where

6. Fuzzy sets - Basic Definitions6 Other approaches to denote fuzzy sets 1. Solely state its membership function. 2. or

7. Fuzzy sets - Basic Definitions7 Example 3 =“integers close to 10” =“integers close to 10” =0.1/7+0.5/8+0.8/9+1/10+0.8/11+0.5/1 2+0.2/13

8. Fuzzy sets - Basic Definitions8 Example 4 =“real numbers close to 10” =“real numbers close to 10”

9. Fuzzy sets - Basic Definitions9 Normal fuzzy set If If the fuzzy set is called normal.

10. Fuzzy sets - Basic Definitions10 Supremum and Infimum For any set of real numbers R that is bounded above, a real number r is called the supremum of R iff For any set of real numbers R that is bounded above, a real number r is called the supremum of R iff r is an upper bound of R r is an upper bound of R no number less than r is an upper bound of R no number less than r is an upper bound of R r=sup R r=sup R For any set of real numbers R that is bounded below, a real number s is called the infimum of R iff For any set of real numbers R that is bounded below, a real number s is called the infimum of R iff s is a lower bound of R s is a lower bound of R no number greater than s is a lower bound of R no number greater than s is a lower bound of R s=inf R s=inf R

11. Fuzzy sets - Basic Definitions11 範例 1 設 X={a,b,c,d} ，給定一 偏序集 (A ， ≤) ，令 ≤={(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (b,d), (a,d)} ，則 c,d 為 A 的上界， 但沒有上確界， a 是 A 的 下界也是 A 的下確界 (infA=a) 。 設 X={a,b,c,d} ，給定一 偏序集 (A ， ≤) ，令 ≤={(a,a), (b,b), (c,c), (d,d), (a,b), (a,c), (b,d), (a,d)} ，則 c,d 為 A 的上界， 但沒有上確界， a 是 A 的 下界也是 A 的下確界 (infA=a) 。 a b d c Hasse diagram Maximal element First and Minimal element

12. Fuzzy sets - Basic Definitions12 範例 2 設 X={a,b,c,d} ，給定一偏序 集 (A ， ≤) ，設 ≤={(a,a), (b,b), (c,c), (d,d), (a,c), (a,d), (b,c), (b,d)} ，令 H={c,d} ，則 H 沒有上界， 且 a,b 都是 H 的下界。令 K={a,b,d} ，則 d 是 K 的上 確界 (supK=d) ，但 K 沒 有下界。 設 X={a,b,c,d} ，給定一偏序 集 (A ， ≤) ，設 ≤={(a,a), (b,b), (c,c), (d,d), (a,c), (a,d), (b,c), (b,d)} ，令 H={c,d} ，則 H 沒有上界， 且 a,b 都是 H 的下界。令 K={a,b,d} ，則 d 是 K 的上 確界 (supK=d) ，但 K 沒 有下界。 a b c d Hasse diagram

13. Fuzzy sets - Basic Definitions13 Support The support of a fuzzy set,S( ), is the crisp set of all x  X such that The support of a fuzzy set,S( ), is the crisp set of all x  X such that

14. Fuzzy sets - Basic Definitions14 Example 1 “comfortable type of house for a 4- person family “may be described as “comfortable type of house for a 4- person family “may be described as ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

15. Fuzzy sets - Basic Definitions15 Support of Example 1 S( )={1,2,3,4,5,6} S( )={1,2,3,4,5,6}

16. Fuzzy sets - Basic Definitions16 α - level set (α- cut) The crisp set of elements that belong to fuzzy set at least to the degree α. The crisp set of elements that belong to fuzzy set at least to the degree α. strong α-level set strong α-cut is called “strong α-level set” or “strong α-cut”.

17. Fuzzy sets - Basic Definitions17 Example 1 “comfortable type of house for a 4- person family “may be described as “comfortable type of house for a 4- person family “may be described as ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

18. Fuzzy sets - Basic Definitions18 α cut of Example 1 ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} A 0.2 ={1,2,3,4,5,6} A 0.5 ={2,3,4,5} A ’ 0.8 ={4} A 0.8 ={3,4} A 1 ={4}

19. Fuzzy sets - Basic Definitions19 Convex crisp set A in  n For every pair of points r=(r i |i  N n ) and s=(s i |i  N n ) in A and every real number λ  [0,1], the point t=(λr i +(1-λ)s i |i  N n ) is also in A. For every pair of points r=(r i |i  N n ) and s=(s i |i  N n ) in A and every real number λ  [0,1], the point t=(λr i +(1-λ)s i |i  N n ) is also in A.

20. Fuzzy sets - Basic Definitions20 Convex fuzzy set A fuzzy set is convex if A fuzzy set is convex if

21. Fuzzy sets - Basic Definitions21 Cardinality | | For a finite fuzzy set Is called the relative cardinality of

22. Fuzzy sets - Basic Definitions22 Example 1 “comfortable type of house for a 4- person family “may be described as “comfortable type of house for a 4- person family “may be described as ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)}

23. Fuzzy sets - Basic Definitions23 Cardinality of example 1 ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} X={1,2,3,4,5,6,7,8,9,10} | |=0.2+0.5+0.8+1+0.7+0.3=3.5 || ||=3.5/10=0.35

24. Fuzzy sets - Basic Definitions24 Basic set-Theoretic operations (standard fuzzy set operations) Standard complement Standard complement Standard intersection Standard intersection Standard union Standard union

25. Fuzzy sets - Basic Definitions25 Standard complement The membership function of the complement of a fuzzy set The membership function of the complement of a fuzzy set

26. Fuzzy sets - Basic Definitions26 Standard intersection The membership function of the intersection The membership function of the intersection

27. Fuzzy sets - Basic Definitions27 Standard union The membership function of the union The membership function of the union

28. Fuzzy sets - Basic Definitions28 Standard fuzzy set operations of example 1 1 ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.3)} =“comfortable type of house for a 4- person-family” ={(3,0.2), (4,0.4), (5,0.6), (6,0.8), (7,1), (8,1),(9,1),(10,1)} =“large type of house”

29. Fuzzy sets - Basic Definitions29 Standard fuzzy set operations of example 1 2 ={(1,1),(2,1),(3,0.8),(4,0.6),(5,0.4),(6,0.2)} ={(3,0.2),(4,0.4),(5,0.6),(6,0.3)} ={(1,0.2), (2,0.5), (3,0.8), (4,1), (5,0.7), (6,0.8),(7,1),(8,1),(9,1),(10,1)}