Complex Fuzzy Sets Towards New Foundations Hung T Nguyen Mathematics New Mexico State University Las Cruces NM USA hunguyennmsuedu Abraham Kandel Computer Science Engineering University of South Florida Tampa FL USA Vladik Kreinovich Computer Science UniversityofTexas El Paso TX USA vladikcsutepedu Abstr actUncertainty of complexvalued physical Then for each the cut for the real part x is an quantities z x iy can be describ ed by complex interval x x the cut for the imaginary part fuzzy sets Such sets can be describ ed by member is also an interval y y and hence the cut for ship functions x y which map the universe of dis the resulting D memb ership function is a rectangular course complex plane into the interval The b ox x x y y The b oundary of problem with this description is that it is dicult to this b ox consists of two straight line segments which are directly translate into words from natural language parallel to the x axis and of two straight line segments To makethis translation easier several authors have which are parallel to the y axis prop osed to use instead of a single memb ership func problems eg when the analyzed In some practical tion for describing the complex number several mem complex fuzzy number z is the result of applying an exp o b ership functions which describ e dierent realvalued nential function to some other complex numb er its cuts characteristics of this numbers such as its real part mayhave a more complicated shap e than a rectangle its imaginary part its absolute value etc The quality Some such situations can b e describ ed by using the fact of this new description strongly dep ends on the choice that in many practical problems it is more convenientto of these realvalued functions so it is imp ortant to represent a complex numb er not in the form z x iy cho ose them optimally In this pap er we formulate which corresp onds to Cartesian co ordinates in the plane the problem of optimal choice of these functions and x y but in the form z expi which corresp onds show that for all reasonable optimality criteria the to p olar co ordinates in this plane For such practical level sets of optimal functions are straight lines and problems A Kaufmann and M Gupta prop osed in to circles This theoretical result is in good accordance use a goniometric representation in which a complex fuzzy with our numerical exp eriments according to which numb er is represented by a pair of real fuzzy numbers such functions indeed lead to a good description of and with membership functions and In complex fuzzy sets this approach for every complex value x iy the degree x y with which this complex value is p ossible can b e Many practical problems lead to complex fuzzy dened as sets Manyphysical quantities are complexvalued wave x y min function in quantum mechanics complex amplitude and p imp edance in electrical engineering etc where x y and arctanyx are the values In all these problems exp ert uncertainty means that we of p olar co ordinates of the p oint whose Cartesian co ordi do not know the exact value of the corresp onding complex nates are x and y For this representation the cut is a number instead we have a fuzzy knowledge ab out this set of all vectors for which is b etween and number and the angle is b etween and On a x y From complex fuzzy numb ers of Kaufmann and plane this cut is no longer a rectangle it is a rather b ership function descrip Gupta to Buckleys mem complicated geometric gure which is b ounded bytwora tion In order to describ e a complex number z x iy dial straight line segments corresp onding to we must describ e two real numb ers its real part x and and and bytw o circular segments corresp ond its imaginary part y Thus a natural idea is to repre ing to and sent a complex fuzzy number by describing two real fuzzy Some complex fuzzy numbers haveeven more compli numbers x and y see eg characterized by the cor cated cuts or in other words memb ership functions resp onding memb ership functions x and y In which cannot be describ ed by the expressions and this approach for every complex value x iy ie for ev To describ e such complex fuzzy numb ers it is nat ery pair x y the degree x y with which this complex ural to use a general memb ership function x y which value is p ossible can b e dened as maps a complex plane C into the interval This ap proachwas sketched in the ab ovementioned b o ok and x y min x y thoroughly develop ed in for the latest overview see It turns out that in many practical problems it is use eg ful to use three or four dierent characteristics For ex ample we can combine Cartesian and p olar ones into Why memb ership function description is some a single characteristic set with t f x y x p times not sucient From the purely mathematical t f x y y t f x y x y and viewp oint this very general approach in which use a gen t f x y arctanyx In this case the cut is eral D memb ership function is very natural However an intersection of a rectangle corresp onding to and a there is one problem with this approach Amembership segment corresp onding to ie a set whose b oundary function is not something which is natural for a human consists partly of straight line circular arcs partly of ra to understand and to use It was invented as a way of dial straight line segments and partly of segments which representing human fuzzy knowledge in a language which are parallel to x or y axes is understandable for a computer From this viewp oint after we get the desired memb ership function we must Which realvalued characteristics of complex num p erform one more step wemust translate it into the nat b ers should we use in this description of complex ural language fuzzy numb ers As shown in the eciency of the new description in solving practical problems with com This translation is dicult even for real variables how plex numb ers strongly dep ends on the appropriate choice ever forrealnumb ers wehave accumulated a lot of in of the realvalued characteristics which are used to de tuition and we are often able to describ e dierent D scrib e the corresp onding fuzzy set a good choice can memb ership functions by naturallanguage words suchas drastically improve the quality of the result It is there small close to etc Unfortunately complex num imp ortant to nd out which functions are the best fore bers are much less intuitive and there are few terms of here This is the problem that we will b e solving in the natural language which can b e naturally used to describ ed present pap er the knowledge ab out complex numb ers Preliminary step reformulation in terms of sets A new approach to describing complex fuzzy sets The memb ership function t corresp onding to a char f which combines the generality of Buckleys ap acteristic f C R can be describ ed by the extension proach with the intuitiveness of KaufmannGupta principle descriptions Since the original memb ership function t max t C is dicult to interpret directly the au f z f z t thors of prop osed a new approach In this approach to describ e a fuzzy knowledge ab out a complex number Thus to b e able to compute all the values t we do not f weuse instead of single memb ership function which de need to compute know the exact characteristics f t it is scrib es the number z x iy several two or more sucient to b e able to describ e their level sets fz j f z memb ership functions which describ e realvalued quanti tg So instead of cho osing the best characteristics we ties whicharefunctions of this complex numb er suchas can cho ose a family of sets its real part Rez its imaginary part Imz its absolute For the ab ove characteristics f x y these level sets i value jz j its phase etc are straight lines and circles In other words instead of describing a single D mem Of course the more parameters we allow in the descrip b ership function x y we describ e several twoormore tion of a family the more elements this family contains memb ership functions t t corresp onding to and therefore the b etter the representation So the ques k k dierent realvalued characteristics t f x y t tion can b e reformulated as follows for a given number of k f x y of this complex number z x iy In this ap k parameters ie for a given dimension of approximating proach for every complex value x iy the degree x y family of sets which is the b est family In this pap er with which this complex value is p ossible can b e dened we formalize and solve this problem as Formalizing the problem All prop osed families of sets x y min t t k k have analytical or piecewise analytical b oundaries so it is natural to restrict ourselves to such families By where t f x y i i denition when wesay that a piece of a b oundary is an If we use two characteristics t f x y x and alytical we mean that it can b e describ ed by an equation t f x y y then we get complex numb ers of typ e F x y for some analytical function in which cuts are rectangles If we use two characteristics t f x y x y a b x c y d x e x y f y F p x y and t f x y arctanyx then we get complex numb ers of typ e in which cuts are So in order to describ e a family we must describ e the ab ovedescrib ed segments corresp onding class of analytical functions F x y For example the level set f x y x t can b e de Denition By an optimality criterion we mean a scrib ed an equation F x y for an analytical function transitive