Inclusion of the Intuitionistic Fuzzy Sets Based on Some Weak Intuitionistic Fuzzy Implication

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1 LGRIN DEMY OF SIENES YERNETIS ND INFORMTION TEHNOLOGIES Volume No 3 Sofia 0 Inclusion of the Intuitionistic Fuzzy Sets ased on Some Weak Intuitionistic Fuzzy Implication Piotr Dworniczak Department of pplied Mathematics niversity of Economics al. Niepodległości Poznań Poland p.dworniczak@ue.poznan.pl bstract: The definition of the Parametric of Inclusion Degree PID of the intuitionistic fuzzy sets on the basis of the weak intuitionistic fuzzy implication is presented in the paper. The aioms which the inclusion measure must fulfill noticed in recent literature are given. The theorems present which of the aioms are met and which are not met by the newly introduced PID. Since the PID fulfills the basic aioms although some in a soft way it can be an alternative to other more hard measures of inclusion. Keywords: Intuitionistic fuzzy sets set inclusion intuitionistic fuzzy implication.. rief introduction to Intuitionistic Fuzzy Sets IFS In 965 L.. Zadeh defined the fuzzy set as Z { Z : } where Z is the value of function Z : [0 ] on a universe called the membership function. The family of all fuzzy sets on the universe we denote as FS. In 983 K. tanassov defined some generalization of Zadeh s fuzzy sets. Definition. Intuitionistic fuzzy set on a universe is understood as { : }

2 where and are functions from to a closed interval [0 ] and for every holds. The values and are respectively the degree of membership and the degree of non-membership of element to the set. We would understand as a couple. nlike in classical fuzzy sets the values of and are independent on each other omitting. The family of all intuitionistic fuzzy sets on we denote as IFS. s classic fuzzy sets are associated with fuzzy logic the IFSs are associated with the so called Intuitionistic Fuzzy Logic IFL. In this logic the truth-value of the propositional variable p is given by an ordered couple a b where a b ab [0 ]. This pair will be called an Intuitionistic Fuzzy Value IFV and the set { a b [0 ] : a b } will be denoted by L. The numbers a and b are interpreted as the validity- and non-validity-degree of the variable p. The truthvalue of the variable p we denote by Vp. The variable with a truth-value true in classical logic is denoted by and the variable false by 0. For this variables holds also V 0 and V0 0. Moreover some kind of fuzzy truth-value is distinguished. We call the variable an Intuitionistic Fuzzy Tautology IFT when for V a b holds: a b and similarly an Intuitionistic Fuzzy co-tautology IFcT if a b holds. The IFT is therefore a variable at least as true as false or rather true. For every we can define the value of negation of in the typical form V b a. onsequently the set c { : } is the standard complement of. In intuitionistic fuzzy propositional calculus Intuitionistic Fuzzy Implication is an important operator. Definition. The weak intuitionistic fuzzy implication is a mapping fulfilling for any variable p p p q q q the properties: i if Vp p Vp then Vp q f Vp q i if Vq p Vq then Vp q p Vp q i3 0 q is an IFT i4 p is an IFT i5 0 is an IFcT where p is a fuzzy ordering relation in the set of an intuitionistic fuzzy variable. Definition is given based on the definition of fuzzy implication a c z y ń ski Jayaram [3 p. ]. 3

3 For the variable p and q with Vp a b and Vq c d we denote Vp p Vq if and only if a c and b d. We call this implication weak implication because properties i3 i4 and i5 are also given in the typical strong form V0 q Vp 0 and V 0 0. There are many postulated properties in literature which the fuzzy implication must meet. Significantly fewer properties are given eactly for the intuitionistic fuzzy implication. They are usually etensions of the fuzzy implications aioms. The main ones are given below.. Strict and graded inclusion in intuitionistic fuzzy environment In theory both classical sets and their generalizations as well as suggestions to use the concept of inclusion is very important. In the case of vague sets the inclusion is not determined unambiguously. It is related to the various definitions of multivalued implications different concepts of cardinalities or even with an ambiguous definition of inclusion. In intuitionistic fuzzy environment the problems are similar. In this paper the conditions that the measure of inclusion must fulfill are presented in a brief literature overview. parametric inclusion degree based on the weak intuitionistic fuzzy implication is introduced. In the paper Fuzzy Sets L.. Zadeh gives a basic definition of the inclusion of two fuzzy sets. Denoted by FS inclusion in Zadeh s sense this definition can be written as follows. Definition 3. Let Y Z FS. It is Y FS Z iff : Y Z. This definition allows the stating of a kind of absolute strict inclusion. Therefore in this case the varying degrees of inclusion relationships are not highlighted. So other different definitions of the inclusion with an inclusion degree in the interval [0 ] are considered. Formally this degree is a value of the mapping I FS : FS FS [0 ] and must meet certain conditions. n etension of this idea is the definition of the IFSs inclusion. Definition 4. Let IFS. It is IFS iff : and. It is easy to notice that the inclusion stated in this way is also rigid. It allows us to respond negatively to both the inclusion of sets { 0: } and { 0 : } and sets with identical membership function and nonmembership function different only at one point 0 in which 0 < 0. Due to the doubts depicted in the eample above different definitions of inclusions and inclusions degree are also introduced for IFSs. sing the operators and given by tanassov as { : } and { : } we can consider also a weaker inclusion in the form 4

4 iff : or iff :. We denote that these two inclusions are really the crisp inclusions of fuzzy sets because FS. nother generalization is the proposal of the designate of the inclusion degree using the mapping I IFS : IFS IFS [0 ] with real values or ID IFS : IFS IFS L with intuitionistic fuzzy values. While defining the degree of inclusions it is most often considered what set of aioms must be adopted for function I IFS and what for ID IFS so that their values could be considered as inclusion degrees. The aioms proposed are usually an etension of the aioms known for fuzzy sets. In various sources there is no compliance even with respect to their canon. sually the aioms proposed are a consequence of the approach given by S i n h a and D o u g h e r t y [4] and also independently by K i t a i n i k [0]. The analysis of the work of Sinha Dougherty and Kitainik is contained in [6]. o r n e l i s and K e r r e [7] etend the approach given also in [6]. ccording to them an intuitionistic fuzzy inclusion measure Inc is an IFS IFS L mapping fulfilling: K Inc Inc c c ; K Inc INFInc Inc where denotes the classical intersection with INF INF def min ma ; K 3 Inc IncP P where P is a IFS IFS mapping defined for as P p and p is a permutation of ; K 4a Inc 0 iff IFS ; K 4b Inc 0 iff : 0 and 0 ; K 4c if FS then Inc D where D { a b L: ab}. The properties K -K 4 are called contrapositivity distributivity symmetry and faithfulness respectively. The property K 4c is called a heritage condition. V l a c h o s and S e r g i a d i s [5] and later X i e H a n and M i [6] state that the subsethood measure or the inclusion measure Inc: IFS IFS [0 ] must fulfill the following properties: VS Inc iff IFS ; VS if c IFS then Inc c 0 iff IFS where IFS { 0: }; VS 3a if IFS IFS then Inc Inc ; 5

5 VS 3b if IFS then Inc Inc. L u o and Y u [] proposed the inclusion degree function in the form Inc: IFS IFS [0 ] satisfying: LY if IFS then Inc ; LY Inc IFS IFS 0 where IFS { 0 : }; LY 3 if IFS IFS then Inc mininc Inc. Q u i a n L i a n g and D a n g [3] considering the inclusion not directly IFSs have established that the degree for the set included in set must be a number from the closed interval [0 ] fulfilling properties LY and VS 3b only. In a recent publication G r z e g o r z e w s k i [9] gives a new approach to the inclusion degree defining the possible and necessary inclusion degree of IFSs but he gives the aioms only for FS called mapping Inc: FS FS [0 ] an inclusion indicator or subsethood measure. The indicator must fulfill the aioms equivalent to K 4a K 4b K and K and the fifth aiom: Inc IncΦ Φ where Φ: FS FS is a mapping defined by Φ φ with φ denoting a function φ:. Let us note that this aiom is strange if φ can be any function. onsidering for eample {t s} and {0/s /t} and {0/s 0/t} if K 4a must be satisfied then Inc is obtained. However taking for eample a fied function φ s we obtain Φ {0/t 0/s} Φ and further IncΦ Φ Inc. Maybe some additional conditions must be imposed on function φ probably as in K 3. Later in this paper this aiom will not be considered. The inclusion degree must be the measure of orming about to what etent how much the set is contained in. Otherwise this degree can be understood as a measure related to the classical definition of the inclusion of in which is as follows: iff : class where class denoted the classical implication. The quantifier for all means that the implication must be true for all elements of. If the values of the implications are the IF values then all of them would have the value 0. This is a very sharp assumption. That is why in the fuzzy environment we usually define the inclusion of sets not in the form yes/no but yes to some degree. If we epand the classical definition recognizing the quantifier for all as the conjunction of all implications for we will get the basic method for calculating the inclusion degree in the form based on idea of a n d l e r and K o h o u t [4]: where Inc INF V 6

6 INF a b def a b. We note that an inclusion degree of the IFSs could be defined also in other forms based on the cardinality of the appropriate sets or on the basis of their distance. 3. The parametric inclusion degree Lemma. The parametric intuitionistic logical connective with a truth-value: b c a d V y where V a b Vy c d and R is a Weak Intuitionistic Fuzzy Implication fulfilling Definition see: [8]. Definition 5. Let IFS. We call the value PID INF V L a Parametric Inclusion Degree PID of into. Remarks Remark. If IFS and IFS then PID is an IFT and PID D is an IFcT. Remark. PID 0 iff and IFS and IFS. Remark 3. PID is an IFT but PID 0. Remark 4. There does not eist any for which PID 0. The following theorems provide fulfillment of the aioms given in the references by PID. We note that the aioms VS and LY relate to the measures of the inclusion with values from the interval [0 ]. In the net theorems we consider the adequate aioms with values in L. Theorem. The Parametric Inclusion Degree fulfills the aioms: a PID K : PID PID c c ; b PID K : PID INFPID PID ; c PID 3 K 3: PID PIDP P where P is a IFS IFS mapping defined for as P p where p is a permutation of ; d PID 4 K 4c: if FS then PID D { a b L: ab}; e PID 5 VS 3a: if IFS IFS then PID p PID ; D 7

7 8 f PID 6 VS 3b: if IFS then PID p PID ; g PID 7 LY 3: if IFS IFS then PID p INFPID PID. P r o o f: a PID INF V INF INF V c c PID c c. b PID INF V INF V INF INF min ma INF min ma INF INF INF INF INF INFPID PID. c PID y INF Vy y p INF Vp p INF Vp p PIDP P. d If FS then and. Therefore PID INF

8 9 * D. The equality * holds because for f[0 ] the property { f} { f} is valid. e IFS IFS means : and :. y definition PID INF and PID INF. ecause : then and because : then therefore PID p PID. f nalogous to e. g Readily apparent from e and f Theorem. The Parametric Inclusion Degree does not fulfill the aioms: a K 4a VS but holds PID 8a: if PID 0 then IFS ; PID 8b: if IFS then PID is an IFT; b K 4b but holds PID 9a: if PID 0 then : 0 and 0 ; PID 9b: if : 0 and 0 then PID is an IFcT; c VS ; d LY ; e LY. P r o o f: a PID 0 iff INF 0

9 0 iff : and 0 iff : and. ecause then and 0. If PID 0 then and : and 0. So it is obtained : 0 and 0 therefore IFS. In the other direction let IFS this means : and. The inequality is equivalent to and this holds because : 0 and 0. In the second case if it were otherwise there would eist such that >0 i. e. >. ut from the assumption therefore the sum > which is impossible. s a result PID is an IFT. b ased on Remark 4 property PID 9a is formally fulfilled. The property PID 9b is fulfilled because if : 0 and 0 then and and because so PID D is an IFcT. c Readily apparent from Definition 5 and Remark 4. d Readily apparent from Theorem a and PID 8b. e Readily apparent from Definition 5 and Remark 4 Remarks Remark 5. If IFS then PID f PID. P r o o f: Let IFS. y definition PID PID. Since : and and therefore and similarly.

10 Finally PID f PID Remark 6. The property PID 5 can be etended to: PID 5a: if IFS then PID p PID. Remark 7. The property PID 9b can be etended to the form PID 9c: if 0 : score 0 score 0 then PID is an IFcT where score :. Namely it is 0 0 * 0 0 and inequality * holds therefore from the assumption 0 : which is equivalent to onclusion The definition of the parametric degree of inclusion of intuitionistic fuzzy sets on the basis of weak intuitionistic implication is presented in the paper. The aioms which must fulfill the inclusion measure noticed in recent literature are given. The theorems indicate which of the aioms are met and which are not met by the newly introduced inclusion degree PID. Since PID fulfills the basic aioms although some of them in a soft way it can be an alternative to other more hard measures of inclusion. Some properties may be difficult to accept for eample the contained in Remark 3 property PID 0. We note however that the properties of PID allow its use in some applications for eample to determine the optimal solution based on the level of inclusion of the IFS describing the solution and IFSs describing the best or worst solution. References. t a n a s s o v K. T. Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems t a n a s s o v K. T. New Operations Defined Over the Intuitionistic Fuzzy Sets. Fuzzy Sets and Systems a c z y ń s k i M.. J a y a r a m. Fuzzy Implications. erlin Springer a n d l e r W. L. J. K o h o u t. Fuzzy Power Sets and Fuzzy Implication Operators. Fuzzy Sets and Systems o s c P. O. P i v e r t. bout pproimate Inclusion and its iomatization. Fuzzy Sets and Systems o r n e l i s.. V a n d e r D o n c k E. K e r r e. Sinha-Dougherty pproach to the Fuzzification of Set Inclusion Revisited. Fuzzy Sets and Systems o r n e l i s. E. K e r r e. Inclusion Measures in Intuitionistic Fuzzy Set Theory. In: T. D. Nielsen N. L. Zhang Eds. ESQR 003 LNI 7. erlin Springer D w o r n i c z a k P. On One lass of Intuitionistic Fuzzy Implications. ybernetics and Information Technologies Vol No 4 3-.

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