L. A. Zadeh: Fuzzy Sets. (1965) A review

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1 POSSIBILISTIC INFORMATION: A Tutorial L. A. Zadeh: Fuzzy Sets. (1965) A review George J. Klir Petr Osička State University of New York (SUNY) Binghamton, New York 13902, USA gklir@binghamton.edu Palacky University, Olomouc, Czech Republic prepared for International Centre for Information and Uncertainty, Palacky University, Olomouc!!!! P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

2 The underlying paper L. A. Zadeh: Fuzzy Sets. Information and Control Outline I Introduction II Definitions III Some properties of,, and complementation IV Algebraic operations on fuzzy sets V Convexity P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

3 Introduction More often than not, the classes of objects encountered in the real physical world do not have precisely defined criteria of membership. For example, the class of animals clearly includes dogs, horses, birds, etc. as its members, and clearly excludes such objects as rocks, fluids, plants, etc. However, such objects as starfish, bacteria, etc. have an ambiguous status with respect to the class fo animals. (... ) Clearly, [examples]... or, the class of tall men, do not constitute classes or sets in the usual mathematical sense of these terms. Yet, the fact remains that such imprecisely defined classes play an important role in human thinking,... P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

4 Introduction The concept in question is that of fuzzy set, that is a class with a continuum of grades of membership. (... )... the notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets... (... ) Essentialy, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply defined criteria of class membership rather than the presence of random variables. P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

5 Definitions Fuzzy Set fuzzy set A in X (space of points) is charaterized by its membership function f A : X [0, 1]. Zadeh knows about the possibility of f A : X P, where P is a poset f A (x) grade of membership of x in A A is an ordinary set iff f A : X {0, 1}... althought the membership function of a fuzzy set has some resemblance to a probability function... there are essential differences between these concepts... In fact, the notion of a fuzzy set is completely nonstatistical in nature. P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

6 Definitions A is empty set iff f A (X) = 0 for all x X. A and B are equal iff f A (x) = f B (x) for all x X. Complement of A is a fuzzy set A such that f A (x) = 1 f A (x). A is contained in B (subset) iff f A (x) f B (x). Union of A and B: f A B (x) = max(f A (x), f B (x)). Union is the smallest fuzzy set containing both A and B. Intersection of A and B: f A B (x) = min(f A (x), f B (x)). Intersection is the largest fuzzy set which is contained in both A and B. P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

7 Definitions... the notion of belonging, which plays a fundamental role in the case of ordinary sets, does no have the same role in the case of fuzzy sets. Thus, it is not meaningful to speak of a point x belonging to a fuzzy set A... Connections to three-valued (Kleene) logic α β : α, β [0, 1] x belongs to A if f A (x) α x does not belong to A if f A (x) β x has an indeterminate status if β < f A (x) < α P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

8 Some properties of,, and complementation Basic Laws (a) (A B) = A B (b) (A B) = A B (c) C (A B) = (C A) (C B) (d) C (A B) = (C A) (C B) Fuzzy sets in X constitute a distributive lattice with a 0 and 1. P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

9 Algebraic operations on fuzzy sets Algebraic product: AB: f AB (x) = f A (x)f B (x); AB A B Algebraic sum: A + B: f A+B (x) = f A (x) + f B (x); meaningful only if f A + f B 1 Absolute difference: A B : f A B (x) = f A (x) f B (x) ; in ordinary case it is relative complement of A B in A B. Convex combination: (A, B; Λ): f (A,B;Λ) (x) = f Λ (x)f A (x) + [1 f Λ ]f B (x); A B (A, B; Λ) A B, P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

10 Algebraic operations on fuzzy sets Fuzzy relation: n-ary fuzzy relation in X: fuzzy set in the product space X X X composition: f B A (x, y) = max v min[f A (x, v), f B (v, y)]; associativity Fuzzy sets induced by mappings T : X Y, B is a fuzzy set in Y; T 1 induces a fuzzy set A in X: f A (x) = f B (y) for T (x) = y the converse problem, A is a fuzzy set in X; T induces a fuzzy set B in Y : f B (y) = max x T 1 (y) f A (x) P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

11 Convexity It is assumed that X is a real Euclidean space E n. Convexity A is convex if its all α-cut are convex. alternative definition: f A [λx 1 + (1 λ)x 2 ] min[f A (x 1 ), f A (x 2 )] A and B are convex then A B is convex Boundedness A is bounded if all α-cuts are bounded, that is there is R(α): x R(α) for all x in α-cut x 0 X is essentially attained: for each ɛ > 0 every spherical neigbourhood of x 0 contains points from the set {x f A (x) max y f A (y) ɛ} preserved by intersection P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

12 Convexity Strict and strong convexity strict: α-cuts are strictly convex. strong: f A [λx 1 + (1 λ)x 2 ] > min[f A (x 1 ), f A (x 2 )] preserved by intersection Shadow of a fuzzy set A fuzzy set in E n shadow on hyperplane H = {x x 1 = 0} is a fuzzy set given by f SH (x 2,..., x n ) = max x1 f A (x 1,..., x n ) for convex fuzzy sets A, B: S H (A) = S H (B) for all H then A = B P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13

13 Convexity Separation of fuzzy sets H hypersurface defined by h(x) = 0, bounded fuzzy sets A and B K H : f A (x) K H for x : h(x) 0 and f B (x) K H for x : h(x) 0. M H = min K H Degree of separation of A and B by H is D = 1 M H Degree of separability of A and B wrt. H λ is D = 1 M, M = min λ M Hλ Theorem Let A, B, be bounded convex fuzzy sets in E n with maximal grades M A, M B. Let M be maximal grade of A B. Then D = 1 M. P. Osička (DAMOL) L. A. Zadeh: Fuzzy Sets. (1965) A review / 13