Fuzzy Systems - Fuzzy Set Theory

1 Fuzzy SystemsFuzzy Set TheoryProf. Dr. Rudolf Kruse Christian University of MagdeburgFaculty of Computer ScienceDepartment of Knowledge Processing and Language EngineeringR. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 2 Outline1. Representation of Fuzzy SetsCantor s TheoryAlpha-cutsProperties based on Alpha-cuts2. Fuzzy Set Operators based on Multi-valued LogicsDefinition of a set By a set we understand every collection made intoa whole of definite, distinct objects of our intuitionor of our thought. (Georg Cantor).For a set in Cantor s sense, the following propertieshold: x6={x}. Ifx XandX Y, thenx/ Y. The Set of all subsets ofXis denoted as 2X. is the empty set and thus very Cantor (1845-1918)R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 22 / 36 Extension to a Fuzzy setling. descriptionmodelall numbers smallerthan 10objective110)[characteristicfunction of asetall numbers almostequal to 10subjective110membershipfunction of a Fuzzy set DefinitionA Fuzzy set ofX6= is a function from the reference setXto theunit interval, :X [0,1].]

2 F(X) represents the set of all fuzzysets ofX, (X)def={ | :X [0,1]}.R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 23 / 36 Vertical RepresentationSo far, Fuzzy sets were described bytheir characteristic/membership function andassigning degree of membership (x) to each elementx is thevertical representationof the corresponding Fuzzy set, expression like aboutm m,d(x) =(1 m xd ,ifm d x m+d0,otherwise,or approximately betweenbandc a,b,c,d(x) = x ab a,ifa x<b1,ifb x cx dc d,ifc<x d0,ifx<aorx> Kruse, C. MoewesFS Fuzzy Set TheoryLecture 24 / 36 Horizontal RepresentationAnother representation is very often applied as follows:For all membership degrees belonging to chosen subset of [0,1],human expert lists elements ofXthat fulfill vague concept of Fuzzy setwith degree .That is thehorizontal representationof Fuzzy sets by their F(X) and [0,1]. Then the sets[ ] ={x X| (x) },[ ] ={x X| (x)> }are called the -cutandstrict -cutof .R. Kruse, C.)

3 MoewesFS Fuzzy Set TheoryLecture 25 / 36A Simple ExampleLetA X, A:X [0,1] A(x) =(1 ifx A,0 otherwise 0< < [ A] =A. Ais called indicator function or characteristic function Kruse, C. MoewesFS Fuzzy Set TheoryLecture 26 / 36An Example01IR [ ]ambLet be triangular function onIRas shown above. -cut of can be constructed by1. drawing horizontal line parallel to x-axis through point(0, ),2. projecting this section onto x-axis.[ ] =([a+ (m a),b (b m)],if 0< 1,IR,if = Kruse, C. MoewesFS Fuzzy Set TheoryLecture 27 / 36 Properties of -cuts IAny Fuzzy set can be described by specifying its is the -cuts are important for application of Fuzzy F(X), [0,1]and [0,1].(a)[ ]0=X ,(b) < = [ ] [ ] ,(c)T : < [ ] = [ ] .R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 28 / 36 Properties of -cuts IITheorem (Representation Theorem)Let F(X). Then (x) = sup [0,1]nmin( , [ ] (x))owhere [ ] (x) =(1,if x [ ] 0, , Fuzzy set can be obtained as upper envelope of its draw -cuts parallel to horizontal axis in height of.)))

4 In applications it is recommended to select finite subsetL [0,1] ofrelevant degrees of must be semantically is, fix level sets of Fuzzy sets to characterize only for these Kruse, C. MoewesFS Fuzzy Set TheoryLecture 29 / 36 System of SetsIn this manner we obtainsystem of setsA= (A ) L,L [0,1],card(L) satisfy consistency conditions for , L:(a) 0 L= A0=X,(fixing of reference set)(b) < = A A .(monotonicity)This induces Fuzzy set A:X [0,1], A(x) = sup L{min( , A (x))}.IfLis not finite but comprises all values [0,1], then must satisfy(c)T : < A =A .(condition for continuity)R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 210 / 36 Representation of Fuzzy SetsDefinitionFL(X) denotes the set of all families (A ) [0,1]of sets that satisfy(a)A0=X,(b) < = A A ,(c)T : < A =A .Any familyA= (A ) [0,1]of sets ofXthat satisfy (a) (b) representsfuzzy set A F(X) with A(x) = sup{ [0,1]|x A }.Vice versa: If there is F(X),then family ([ ] ) [0,1]of -cuts of satisfies (a) (b).

5 R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 211 / 36 Approximately 5 or greater than or equal to 7 An Exemplary Horizontal ViewSuppose thatX= [0,15].An expert choosesL={0, , , ,1}and -cuts: A0= [0,15], [3,15], [4,6] [7,15], [ , ] [7,15], A1={5} [7,15]. family (A ) Lof sets induces upper shown Fuzzy Kruse, C. MoewesFS Fuzzy Set TheoryLecture 212 / 36 Approximately 5 or greater than or equal to 7 An Exemplary Vertical View Ais obtained as upper envelope of the familyAof difference between horizontal and vertical view is horizontal representation is easier to process in , restricting the domain of x-axis to a discrete set is usually Kruse, C. MoewesFS Fuzzy Set TheoryLecture 213 / 36 Horizontal Representation in the sets are usually stored as chain of linear each -level, 6= finite union of closed intervals is stored by their data structure is appropriate for arithmetic Kruse, C. MoewesFS Fuzzy Set TheoryLecture 214 / 36 Support and Core of a Fuzzy SetDefinitionThesupport S( ) of a Fuzzy set F(X) is the crisp set thatcontains all elements ofXthat have nonzero membership.

6 FormallyS( ) = [ ]0={x X| (x)>0}.DefinitionThecore C( ) of a Fuzzy set F(X) is the crisp set that containsall elements ofXthat have membership of one. Formally,C( ) = [ ]1={x X| (x) = 1}.R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 215 / 36 Height of a Fuzzy SetDefinitionTheheight h( ) of a Fuzzy set F(X) is the largest membershipgrade obtained by any element in that set. Formally,h( ) = supx X (x).h( ) may also be viewed as supremum of for which [ ] 6= .DefinitionA Fuzzy set is callednormalwhenh( ) = is calledsubnormalwhenh( )< Kruse, C. MoewesFS Fuzzy Set TheoryLecture 216 / 36 Convex Fuzzy Sets IDefinitionLetXbe a vector space. A Fuzzy set F(X) is calledfuzzy convexif its -cuts are convex for all (0,1].The membership function of a convex Fuzzy setis not classical definition: The membership functions are Kruse, C. MoewesFS Fuzzy Set TheoryLecture 217 / 36 Fuzzy NumbersDefinition is a Fuzzy number if and only if is normal and [ ] is bounded,closed, and convex (0,1].))

7 Example:The termapproximately x0is often described by a parametrized classof membership functions, 1(x) = max{0,1 c1|x x0|},c1>0, 2(x) = exp( c2kx x0kp),c2>0,p Kruse, C. MoewesFS Fuzzy Set TheoryLecture 218 / 36 Convex Fuzzy Sets II01IR x1x2 TheoremA Fuzzy set F(IR)is convex if and only if ( x1+ (1 )x2) min{ (x1), (x2)}for all x1,x2 IRand all [0,1].R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 219 / 36 Fuzzy Numbers [ ] = [1,2]if ,[ + ,2) if 0< < ,IRif = 0 Upper semi-continuous functions are most many applications ( control) the class of the functions andtheir exact parameters have a limited influence on the local monotonicity of the functions is really other applications ( diagnosis) more precise membershipdegrees are Kruse, C. MoewesFS Fuzzy Set TheoryLecture 220 / 36 Set Operators.. are defined by using traditional logics operatorLetXbe universe of discourse (universal set):A B={x X|x A x B}A B={x X|x A x B}Ac={x X|x/ A}={x X| (x A)}A Bif and only if (x A) (x B) for allx XOne idea to define Fuzzy set operators: use Fuzzy Kruse, C.]

8 MoewesFS Fuzzy Set TheoryLecture 221 / 36 The Traditional or Aristotlelian LogicWhat is logic about? Different schools speak different languages!There are raditional, linguistic,psychological, epistemological andmathematical logic has been founded byAristotle (384-322 ).Aristotlelian logic can be seen asformal approach to human s still used today in ArtificialIntelligence for knowledgerepresentation and reasoning of The School of Athens by R. Sanzio (1509)showing Plato (left) and his student Aristotle (right).R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 222 / 36 Outline1. Representation of Fuzzy Sets2. Fuzzy Set Operators based on Multi-valued LogicsBoolean Algebran-valued LogicsFuzzy LogicClassical Logic: An OverviewLogic studies methods/principles logic deals withpropositions(eithertrueorfalse).Thep ropositional logichandles combination oflogical idea: how to expressn-ary logic functions withlogic primitives, , , , .A set of logic primitives iscompleteif any logic function can becomposed by a finite number of these primitives, { , , },{ , },{ , },{ }(NOR),{|}(NAND)(this was also discussed during the 1st exercise).

9 R. Kruse, C. MoewesFS Fuzzy Set TheoryLecture 223 / 36 Inference RulesWhen a variable represented by logical formula is:truefor all possible truth values, is calledtautology,falsefor all possible truth values, is forms of tautologies exist to performdeductive inferenceThey are calledinference rules:(a (a b)) b(modus ponens)( b (a b)) a(modus tollens)((a b) (b c)) (a c)(hypothetical syllogism) ponens: given two true propositionsaanda b(premises), truth of propositionb(conclusion) can be tautology remains a tautology when any of its variables isreplaced with an arbitrary logic Kruse, C. MoewesFS Fuzzy Set TheoryLecture 224 / 36 Boolean AlgebraThe propositional logic based on finite set of logic variables isisomorphic tofinite set of these Systems are isomorphic to a finiteBoolean algebraon a setBis defined as quadrupleB= (B,+, ,)whereBhas at least two elements (bounds) 0 and 1, + and arebinary operators onB, andis a unary operator onBfor which thefollowing properties Kruse, C.

10 MoewesFS Fuzzy Set TheoryLecture 225 / 36 Properties of Boolean Algebras I(B1) Idempotencea+a=aa a=a(B2) Commutativitya+b=b+aa b=b a(B3) Associativity(a+b) +c=a+ (b+c)(a b) c=a (b c)(B4) Absorptiona+ (a b) =aa (a+b) =a(B5) Distributivitya (b+c) = (a b) + (a c)a+ (b c) = (a+b) (a+c)(B6) Universal Boundsa+ 0 =a,a+ 1 = 1a 1 =a,a 0 = 0(B7) Complementarya+a= 1a a= 0(B8) Involutiona=a(B9) Dualizationa+b=a ba b=a+bProperties (B1)-(B4) are common to everylattice, Boolean algebra is a distributive (B5), bounded (B6), andcomplemented (B7)-(B9) lattice, Boolean algebra can be characterized by a partial ordering ona set, a bifa b=aor, alternatively, ifa+b= Kruse, C. MoewesFS Fuzzy Set TheoryLecture 226 / 36 Set Theory , Boolean Algebra, Propositional LogicEvery theorem in one Theory has a counterpart in each other can be obtained applying the following substitutions:MeaningSet Theory Boolean Algebra Prop. Logicvalues2 XBL(V) meet / and join / or + complement / not c identity elementX11zero element 00partial order power set 2X, set of logic variablesV, set of all combinationsL(V) oftruth values ofVR.