1. Ch. 2 of Neuro-Fuzzy and Soft Computing

2. Fuzzy Sets: Outline Introduction Basic definitions and terminology Set-theoretic operations MF formulation and parameterization MFs of one and two dimensions Derivatives of parameterized MFs More on fuzzy union, intersection, and complement Fuzzy complement Fuzzy intersection and union Parameterized T-norm and T-conorm

3. Fuzzy Sets Sets with fuzzy boundaries A = Set of tall people Heights 5’10’’ 1.0 Crisp set A Membership function Heights 5’10’’6’2’’.5.9 Fuzzy set A 1.0

4. Membership Functions (MFs) Characteristics of MFs: Subjective measures Not probability functions MFs Heights 5’10’’.5.8.1 “tall” in Asia “tall” in the US “tall” in NBA

5. Fuzzy Sets Formal definition: A fuzzy set A in X is expressed as a set of ordered pairs: Universe or universe of discourse Fuzzy set Membership function (MF) A fuzzy set is totally characterized by a membership function (MF).

6. Fuzzy Sets with Discrete Universes Fuzzy set C = “desirable city to live in” X = {SF, Boston, LA} (discrete and nonordered) C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)} Fuzzy set A = “sensible number of children” X = {0, 1, 2, 3, 4, 5, 6} (discrete universe) A = {(0,.1), (1,.3), (2,.7), (3, 1), (4,.6), (5,.2), (6,.1)}

7. Fuzzy Sets with Cont. Universes Fuzzy set B = “about 50 years old” X = Set of positive real numbers (continuous) B = {(x,  B (x)) | x in X}

8. Alternative Notation A fuzzy set A can be alternatively denoted as follows: X is discrete X is continuous Note that  and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.

9. Fuzzy Partition Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”: lingmf.m

10. More Definitions Support Core Normality Crossover points Fuzzy singleton  -cut, strong  -cut Convexity Fuzzy numbers Bandwidth Symmetricity Open left or right, closed

11. Definition Support The support of a fuzzy set A is the set of all points x in X such that : support (A) = Definition 2.3 Core The core of a fuzzy set A is the set of all points x in X such that : core (A) =

12. Definition 2.4 Normality A fuzzy set A is normal if its core is nonempty. In other words, we can always find a point such that. Definition 2.5 Crossover points A crossover point of a fuzzy set A is point at which : crossover (A) =

13. Definition 2.6 The of a fuzzy set A is a crisp set defined by support (A) = Strong are defined similarly: Using the notation for a level set, we can express the support and core of a fuzzy set A as support (A) =

14. Definition 2.7 Fuzzy singleton A fuzzy set whose support is a single point in X with is called a fuzzy singleton. Definition 2.8 Convexity A fuzzy set A is convex if and only if for any and any

15. Convexity of Fuzzy Sets A fuzzy set A is convex if for any in [0, 1], Alternatively, A is convex is all its  -cuts are convex. convexmf.m

16. Definition 2.9 Fuzzy numbers A fuzzy number A is a fuzzy set in the real line (R) that satisfies the conditions for normality and convexity. Most (noncomposite) fuzzy sets used in the literature satisfy the conditions for normality and convexity, so fuzzy numbers are the most basic type of fuzzy sets.

17. Definition 2.10 Bandwidths of normal and convex fuzzy sets For a normal and convex fuzzy set, the bandwidth or width is defined as the distance between the two unique crossover points: width (A)= where

18. Definition 2.11 Symmetry A fuzzy set A is symmetric if its MF is symmetric around a certain point x=c for all.

19. Shapes of MF

20. lingmf.m

21. MF Terminology MF X.5 1 0 Core Crossover points Support  - cut 

22. Set-Theoretic Operations Subset: Complement: Union: Intersection:

23. Set-Theoretic Operations subset.m fuzsetop.m

24. MF Formulation Triangular MF: Trapezoidal MF: Generalized bell MF: Gaussian MF:

25. MF Formulation disp_mf.m

26. MF Formulation Sigmoidal MF: Extensions: Abs. difference of two sig. MF Product of two sig. MF disp_sig.m

27. MF Formulation L-R MF: Example: difflr.m c=65 a=60 b=10 c=25 a=10 b=40

28. Cylindrical Extension Base set ACylindrical Ext. of A cyl_ext.m

29. 2D MF Projection Two-dimensional MF Projection onto X Projection onto Y project.m

30. 2D MFs 2dmf.m

31. Fuzzy Complement General requirements: Boundary: N(0)=1 and N(1) = 0 Monotonicity: N(a) > N(b) if a < b Involution: N(N(a) = a Two types of fuzzy complements: Sugeno’s complement: Yager’s complement:

32. Fuzzy Complement negation.m Sugeno’s complement: Yager’s complement:

33. Fuzzy Intersection: T-norm Basic requirements: Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a Monotonicity: T(a, b) < T(c, d) if a < c and b < d Commutativity: T(a, b) = T(b, a) Associativity: T(a, T(b, c)) = T(T(a, b), c) Four examples (page 37): Minimum: T m (a, b) Algebraic product: T a (a, b) Bounded product: T b (a, b) Drastic product: T d (a, b)

34. Fuzzy Intersection: T-norm Minimum: Algebraic product: Bounded product: Drastic product:

35. T-norm Operator Minimum: T m (a, b) Algebraic product: T a (a, b) Bounded product: T b (a, b) Drastic product: T d (a, b) tnorm.m

36. Fuzzy Union: T-conorm or S-norm Basic requirements: Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a Monotonicity: S(a, b) < S(c, d) if a < c and b < d Commutativity: S(a, b) = S(b, a) Associativity: S(a, S(b, c)) = S(S(a, b), c) Four examples (page 38): Maximum: S m (a, b) Algebraic sum: S a (a, b) Bounded sum: S b (a, b) Drastic sum: S d (a, b)

37. Fuzzy Union: T-conorm or S-norm Maximum: Algebraic sum: Bounded sum: Drastic sum:

38. T-conorm or S-norm tconorm.m Maximum: S m (a, b) Algebraic sum: S a (a, b) Bounded sum: S b (a, b) Drastic sum: S d (a, b)

39. Generalized DeMorgan’s Law T-norms and T-conorms are duals which support the generalization of DeMorgan’s law: T(a, b) = N(S(N(a), N(b))) S(a, b) = N(T(N(a), N(b))) T m (a, b) T a (a, b) T b (a, b) T d (a, b) S m (a, b) S a (a, b) S b (a, b) S d (a, b)

40. Parameterized T-norm and S-norm Parameterized T-norms and dual T-conorms have been proposed by several researchers: Yager Schweizer and Sklar Dubois and Prade Hamacher Frank Sugeno Dombi

41. Homework 1 (20%) Exercise 3 of Chapter 2. (You don't need to give proof to every identity. Just list those identities that are not satisfied any more, and give simple examples or explanations why they are not satisfied.) (20%) Exercise 4 of Chapter 2. (You don't need to give proof to every identity. Just list those identities that are not satisfied any more, and give simple examples or explanations why they are not satisfied.) (20%) Exercise 9 of Chapter 2. (20%) Exercise 20 of Chapter 2. Prove that the two equations in (2.50) (page 40) for the generalized DeMorgan's law are equivalent if the complement operator fulfills the involution requirement in Equation (2.38), page 35. (20%) (a) and (c) of Exercise 21, Chapter 2. (You can use the result from the previous question of this homework.)