Where are we? Operations on fuzzy sets (cont.) Fuzzy Logic. Motivation. Crisp and fuzzy sets. Examples

Transcription

1 Operations on fuzzy sets (cont.) G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, chapters -5 Where are we? Motivation Crisp and fuzzy sets alpha-cuts, support, core, height, convexity fuzzy numbers scalar and fuzzy cardinality basic operations of fuzzy sets t-norms and aggregation operators Cartesian product and fuzzy relations properties of set operators geometric representation subsethood Fuzzy Logic n-valued logics fuzzy logics linguistic variables fuzzy propositions linguistic hedges Fuzzy quantifiers Fuzzy inference Examples

2 Cartesian product A B = {( i( μ ( x), μ ( y)),( x, y)) : x U, y V} A B Crisp relation (R A B) A a a 2 a 3 b b 2 b 3 b 4 B a 4 b 5 M R = R ar b a R b a 3 2R b5 ( a, b ),( a, b ),( a, b ) = ( a3, b),( a3, b4),( a4, b2) a 3R b a 3R b a 4 4R b2 2

3 Fuzzy relation (R A B) A x a 2 a 3 a b b 2 b 3 b 4 b 5 B Fuzzy relation, example If both X and Y consists of finite, countable elements, then µ R (x,y) can be represented by a (fuzzy) matrix. Let X = {a, b, c} and the fuzzy relation R on X X: R = 0.2/(a,a) + /(a,b) + 0.4/(a,c) + 0.6/(b,b) + 0.3/(b,c) + /(c,b) + 0.8/(c,c) or 0.2 R = b a c 0.8 3

4 Fundamental properties of set operators Fuzzy sets, geometric representation Let X={x,x 2,..,x n } with X = n. A fuzzy subsets of X can be represented as a point of the n-dimensional hypercube. The vertices represent crisp sets. Example for U = {x,x 2 } (0,) (,) μ A (x )=0.3 μ A (x 2 )=0.4 x 2 U ={x,x 2 } x 2 A(0.3,0.4) (0,0) (,0) Φ x x 4

5 Subsethood Given any pair of fuzzy sets defined on X, A is a subset of B (A B) iff for all x X μ A (x) μ B (x) For any crisp sets A, B A B iff A B = A and A B = B for any Subsethood, degree of Given any pair of fuzzy sets defined on a finite set X, the degree of subsethood, S(A,B), can be given by S(A,B) = A B / A with 0 S(A, B) 5

6 Fuzzy Logic G. J. Klir, B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, chapters -5 Fuzzy Logic: Narrow and broad sense Narrow sense: is symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, etc.; both propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness. Broad Sense: (older, better known, heavily applied but not asking deep logical questions) ( ) It is one of the techniques of soft-computing, i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and sufficiently good solutions. 6

7 Three-valued logics, example of ab Łukasiewicz Bochvar Kleene Heyting Reichenbach ½ 0½½ ½½½½ 0½½ 0½0 0½½ ½0 0½½½ ½½½½ 0½½½ 0½00 0½½½ ½½ ½½ ½½½½ ½½½½ ½½ ½½ ½ ½½ ½½½½ ½½ ½½ ½½ ½ ½½½ ½½½½ ½½½ ½½½ ½½½ Multivalued logics n-valued logics The set T n of truth values of an n-valued logic is thus defined as T n = 0 = 0 2 n 2 n,,, K, =. n n n n n These values can be interpreted as degrees of truth. 7

8 Multivalued logics Lukasiewicz uses truth values in T n and defines the primitives by the following equations: a = a a b = min(, + b a) a b = ( a b) b, a b = a b, a b = ( a b) ( b a) = a b Linguistic variables Values of a linguistic variable are linguistic terms, represented as fuzzy sets, and characterize concepts Membership values Linguistic terms Young Middle-aged Old Age 8

9 Type of fuzzy propositions. Unconditional and unqualified propositions 2. Unconditional and qualified propositions 3. Conditional and unqualified propositions 4. Conditional and qualified propositions Unconditional an unqualified fuzzy propositions The canonical form of a fuzzy propositions of this type, p, is p : V is F, Given a particular value of V (say, v), this value belongs to F with membership grade F(v). This membership grade is then interpreted as the degree of truth, T(p), of proposition p. That is, T ( p) = F( v) 9

10 Unconditional an unqualified fuzzy propositions Unconditional and qualified fuzzy propositions Propositions p of this type are characterized by either p : V is F is S, or p : Proposition{ V is F}is S The degree of truth, T(p), of a truth-qualified proposition p is given for each v V by T ( p) = S( F( v)). Example: "Tina is young is very true." 0

11 Unconditional and qualified fuzzy propositions Example: "Tina is young is very true." Example 2: "Tina is very young is false."