Book of proceedings - The Academic Conference of African Scholar Publications & Research International on Achieving Unprecedented Transformation in a fast- moving World: Agenda for Sub-Sahara Africa. Vol.3 No. 3. 21st May, 2015- University of Abuja, Teaching Hospital Conference Hall, Gwagwalada, Abuja FCT. Vol.5 No.3

DIFFERENCE AND SYMMETRIC DIFFERENCE OPERATIONS DEFINED ON INTUITIONISTIC FUZZY MULTISETS

PAUL AUGUSTINE EJEGWA Department of Mathematics/Statistics/Computer Science, University of Agriculture, P.M.B. 2373, Makurdi Nigeria

Abstract: The concept of intuitionistic fuzzy multisets which is an extension of intuitionistic fuzzy sets is a relative new research area that ushers wide range of applications in real world. We gave a comprehensive note on the fundamentals of intuitionistic fuzzy multisets (IFMSs) which cuts across some foundational definitions, basic operations, Cartesian product, and some laws of algebra. We proposed and defined relative complement and symmetric difference over IFMSs. Thereafter, some theorems and corollaries are given, and proved respectively.

Keywords: algebra, difference, intuitionistic fuzzy sets, intuitionistic fuzzy multisets, operations, symmetric difference.

Introduction Intuitionistic fuzzy set (IFS) introduced by Atanassov [1] as a generalization of fuzzy set proposed earlier by Zadeh [13] received ample attentions in fuzzy community due to its flexibility, applicability and resourcefulness in tackling the issue of vagueness or the representation of imperfect knowledge in Cantor’s set theory. The main advantage of IFSs is their capability to cope with the hesitancy that may exist. This is achieved by incorporating a second function, along with the membership function of the conventional fuzzy sets called non-membership function. Atanassov [3, 4] carried out rigorous research based on the theory and applications of intuitionistic fuzzy sets. Notwithstanding, there are times that each element has different membership values with a corresponding non-membership values. Due to such situations, Shinoj and Sunil [8] introduced IFMSs from the combination of IFSs [2] and fuzzy multisets (FMSs) proposed by Yager [12] and showed it application in medical diagnosis. Obviously, IFMS is a generalized IFS or an extension of FMSs as in [7]. Shinoj and Sunil [10] proposed the accuracy of collaborative robots using the concept of intuitionistic fuzzy multiset approached and also provided an in depth work on IFMS in [9]. Ibrahim and Ejegwa [7] proposed some modal operators on IFMS and gave some proofs based on the concept. Ejegwa and Awolola [17]gave an application of IFMSs in binomial distributions. Some distance measures in line with the ones proposed by Szmidt and Kacprzyk [14, 15] with respect to IFSs were introduced in [16] in terms of IFMSs. In this paper, we will defined relative complement and symmetric difference over IFMS and thereafter, give some theorems and corollaries with proofs. Concept of intuitionistic fuzzy multisets Definition 1 [8]: Let be a nonempty set. An IFMS drawn from is characterized by two functions: “count membership” of denoted as and “count non-membership” of denoted as given respectively by : ⟶ and :⟶ where is the set of all crisp multisets drawn from the unit interval [0,1] s.t. for each ∈ , the membership sequence is defined as a decreasingly ordered sequence of elements in () and it is denoted as ((), (),…,()), where()≥ ()≥…≥ () whereas the corresponding non-membership sequence of elements in () is denoted by ( (), (), …, ()) s.t. 0 ()+  ()1 for every ∈ and = 1, . . ., . This means, an IFMS is defined as; = {,() ,(): x∈X } or A= {, (),  (): ∈ }, for = 1, . . . , . For each IFMS in , π() = 1 – ()−  () is the intuitionistic fuzzy multisets index or hesitation margin of x in . The hesitation margin π()for each = 1, . . ., is the degree of non-determinacy of ∈ , to the set and π() ∈ [0,1]. Similarly, π()as in IFS, is the function that expresses lack of knowledge of whether ∈ or ∉ . In general, an IFMS is given as = {,(),  (),π(): ∈X}, or {,(x),  (x),1 – ()−  (): ∈X}, or {, (x),1−(x ) − π(),π(): ∈ }, or {,1–  ()− π(),  (),π(): ∈X} since ()+  ()+π() =1 for each = 1, . . ., . Definition 2: We define IFMS alternatively. Let be nonempty set. An IFMS drawn from is given as ={µ(),…, µ(),…,  (),…,  (), … : ∈ } where the functions (),  (): → [0,1] define the belongingness degrees and the non-belongingness degrees of in s.t. 0 ()+  ()1 for = 1,… If the sequence of the membership functions and non-membership ( belongingness functions and non-belongingness functions) have only n-terms (i.e. finite), n is called the ‘dimension’ of . Consequently ={µ(),…, µ(),  (),…,  (): ∈ } for = 1,…, . when no ambiguity arises, we define ={µ(),  (): ∈ } for =1,…, . Definition 3 [9]: The length of an element in an IFMS is defined as the cardinality of () or () for which 0≤ () + ()≤1 and it is denoted by (: ) i.e. the length of in for each . Then,

(: ) = ⃒() ⃒ = ⃒()⃒ Definition 4: If and are IFMSs drawn from , then (: , ) = { (: ), (: )} or () = [(: ), (: )] where ()= (: , ) and denotes maximum. Example Consider the set ={, , , } with : (0.3, 0.2), (0.4,0.5), (0.3,0.3), = : (1.0,0.5, 0.5), (0.0,0.5, 0.5), (0.0, 0.0,0.0), : (0.5,0.4,0.3,0.2),(0.4,0.6,0.6,0.7),(0.1,0.0,0.1,0.1) : (0.4), (0.2), (0.4), = : (1.0,0.3, 0.2), (0.0,0.4,0.5), (0.0,0.3,0.3), : (0.2,0.1), (0.7 0.8), (0.1,0.1) Find: (: ), (: ), (: ), (: ), (: ), (: ), (: ), (: ), (: , ), (: , ), (: , ) an d (: , ) respectively. Solution (: ) = 0 i.e. IFMS,(: ) = 2, (: ) = 3, (: ) = 4, (: ) = 1, (: )= 3,(: ) = 2, (: ) = 0 . . IFMS, (: , ) = 1, (: , )= 3, (: , ) = 3 , (: , ) = 4. Note:1. In an IFMS, |() | = | () | for each = 1,2,…, . 2. Whenever =1, an IFMS becomes IFS. 3. IFMS and FMS of the same length have equal cardinality. 4. Whenever the hesitation margin equals zero, an IFMS becomes FMS. Definition 5 (Empty IFMS): If (: ) = | () | = | () | = 0  is an empty IFMS. That is ()=1. Definition 6 (Similar IFMSs): Two IFMSs and are said to be similar or cognate if  () = () or () = () for at least one ( = 1,…, ). Definition 7 (Comparable IFMSs): Two IFMSs and are said to be equal or comparable if () = () and () =() . Definition 8 (Equivalent IFMSs): Two IFMSs and are said to be equivalent to each other i.e. is equivalent to , denoted by ~ if ∃ function : → which is both injection and surjection (i.e. bijection). Then, the function defines a one-to-one correspondence between and . In short, ~ iff (: ) = (: ) provided and are drawn from a nonempty set such that . Definition 9 (Inclusive IFMS): Let and be two IFMSs,   () ≤() and () ≥() for  and = 1,2,…, . Then is a subset of and is a superset of . Definition 10 (Proper Subset): is a proper subset of i.e.  if  and . It means () ≤ () and () ≥() but () and () () for  and = 1,2,…, . Definition 11 (Dominations): An IFMS is dominated by another IFMS (i.e. ≼), if there exist an injection from to . is strictly dominated by (i.e. ≺), if (i) ≼ and (ii) is not equinumerous with . Definition 12 (Relations): Let , and be IFMS in , Then; i. ≼ i.e. is reflexive relation, ii. ≼ and ≼ i.e. symmetric relation, iii. ≼ and ≼≼ i.e. transitive relation iv. ≼ and ≼⇒= i.e. antisymmetric. Corollary 1: For any IFMS and , if ≼ and ≼  ~ i.e. (: ) = (: ). Corollary 2: For any IFMS and , if ≼, ≼ and ≼ and are compatible to each other. The proofs of corollary 1 and 2 are straightforward. Note: If a relation is reflexive, symmetric and transitive, such a relation is called an “equivalence relation”. Operations on intuitionistic fuzzy multisets [8] For any two IFMSs and drawn from , the following operations hold. Let ={,(),  (): ∈ X}and ={,(),  (): ∈X}, for each = 1,2,…, . 1. Complement: ={,(), (): x∈X} 2. Union: ∪ ={, ((), ()), ((),()): } 3. Intersection: ∩ ={, ((), ()), ((),()):} 4. Addition:  ={,()+ ()− ()(), () ():} 5. Multiplication:  ={, ()(), ()+ ()−() ():} Algebraic laws in intuitionistic fuzzy multisets Let , and be IFMSs in , then the following algebra follow: 1. Complementary law: () = 2. Idempotent laws: () ∪ = () ∩ = 3. Commutative laws: () ∪ = ∪ () ∩ = ∩ () =  () =  4. Associative laws: ()( ∪ ) ∪ = ∪ ( ∪ ) () ( ∩ ) ∩ = ∩ ( ∩ ) ()( ) = ( ) ()( ) = ( ) 5. Distributive laws:() ∪ ( ∩ ) = ( ∪ ) ∩ ( ∪ ) () ∩ ( ∪ ) = ( ∩ ) ∪ ( ∩ ) () ( ∪ ) = () ∪ ( ) ()( ∩ ) = ( ) ∩ ( ) ()( ∪ ) = () ∪ ( ) ()( ∩ ) = () ∩ ( ) 6. De Morgan’s laws: () ( ∪ ) = ∩ () ( ∩ ) = ∪ ()( ) = ()( ) =  7. Absorption laws: () ∩ ( ∪ ) = () ∪ ( ∩ ) = Note: Distributive laws hold for both right and left hands.

Theorem 1: Let , , be IFMSs in and ⊆ , then (i) B⊆A (ii)  B⊆A  (iii) ∪B⊆ A∪ (iv) ∩B⊆A∩ . Proof: (i) Given that , , ∈ and ⊆ , it means () ≤ () and () ≥ () for every . If another IFMS ∈ is added to ⊆ and since ⊆ , it is certain that, B⊆A and the result follows. Results of (ii) – (iv) follow from the proof of (i). Theorem 2: Let and be IFMSs in, then (i) ∩ = or ∩ = , (ii) ∪ = or ∪ = iff = . Proof: (i) For , ∈ , it implies that ∩ ∈ . = ⇒ () = ()and() = () ∈ . Since = , from idempotent laws, ∩ = and ∩ = . Result of (ii) follows from (i). Definition 13: We defined Cartesian product by extending its definition from IFS. Let ={, μ(),…,μ(),  (),…,  (): ∈ } and ={,μ(),…,μ(),  (),…, (): ∈ } for = 1,…, , × = ={μ()μ (),  () (): ∈ } for = 1,…, . From Def. 13 and the basic operations, we deduced the following: 1. × = × 2. ( × )× = ×( × ) 3. × ( ∪ ) =( × )∪( × ) 4. × ( ∩ ) =( × )∩( × ) 5. × () =( × )( × ) 6. × () =( × )( × ) The proofs of the above results are directly from the basic operations and Def. 13. Difference operation defined over intuitionistic fuzzy multisets Definition 14: Let be a nonempty s.t. two IFMSs and are in . The difference orrelative complementof and denote by ⃥ or [, ] is defined as ⃥ ={,min (μ(),  ()), max(  () ,μ()): ∈ }. Obviously, for any two IFMSs and , ⃥ is also IFMS. Note: − is also use to denote difference as given in [11]. We adopt − in the course of this research. Theorem 3: Let and be two IFMSs in a nonempty set , then (i) − = ∩ (ii) − = − iff = (iii) − = − . Proof:(i)Let ={μ(),  () : ∈ } and ={,μ(), () : ∈ } for , ∈ and = 1,2,…, , then; − ={,min (μ(),  ()), max(  () ,μ()): ∈ }. But = {,  (), μ(): ∈ },⇒ ∩ ={,min (μ(),  ()), max(  (), μ()): ∈ } since ∩ ={,min (μ(),μ ()), max(  () ,ν()): ∈ }. The result follows. (ii)From Def. 14, − ={,min(μ (),  ()), max(  () ,μ()): ∈ }. If = ⇒μ() = μ () and  ()=ν()∀ ∈ . From this, it is certain that − = − and the result follows. (iii) From Def.14, − ={,min (μ(),  ()), max(  () ,μ()): ∈ }. Given that, = {,  (), μ(): ∈ } and ={,  (), μ(): ∈ }, it implies that, − = {,min ( (), μ()),max(μ (),  ()): ∈ } = − and the result is straightforward. Theorem 4:Let and be two IFMSs in a nonempty set , then; (i) − =⏀ (ii) −⏀= (iii) − ⊆ if ⊆ (iv) − = ⏀ iff = (v) − = iff = ⏀ It is easy to prove the above results. Symmetric difference operation defined over intuitionistic fuzzy multisets Definition 15: Let be a nonempty s.t. two IFMSs and are in . The symmetric difference of and denote by A∆B is defined as

,max  min(μ(),  ()),min μ(),  () , A∆B = min max (),μ() ,max (),μ() : ∈ Note: 1.A∆B is an extension of A−B2. Symmetric difference corresponds to the XOR operation in Boolean Logic 3. Symmetric difference of A and B is variously written as A⦵B,A  B, or A  B as given in [5, 6]. Corollary 3: Whenever = , A  B=B  A∀ , ∈ . Proof is straightforward from the proof of theorem3 (ii). Theorem 5:For IFMSs, , in and ⊆ ⊆ , then we have; (i) B−A⊆ − (ii) B  A⊆  . Proof: (i) Given that , , ∈ and ⊆ ⊆ ⇒ “⊆” is transitive i.e. ⊆ and ⊆ ⇒ ⊆ . Obviously, μ()≤μ()≤μ() and ()≥ ()≥ ()∀ ∈ . Since is the smallest of and , subtracting from both side of ⊆ ⇒B−A⊆ − . The result follows. Since symmetric difference is the extension of relative compliment, the result of (ii) follows. References K. Atanassov, Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, 1983. K. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. K. Atanassov, Intuitionistic fuzzy sets: theory and application, Springer-Verlay (1999). K. Atanassov, On Intuitionistic fuzzy sets, Springer-Verlay (2012). E.J. Borowski, J.M. Borwein, The HarperCollins Dictionary of Mathematics, New York: HarperCollins (1991). J.W. Harris, H. Stocker, Handbook of Mathematics and Computational Science, New York: Springer-Verlag (1998). A. M. Ibrahim, P. A. Ejegwa, Some modal operators on intuitionistic fuzzy multisets, Int. Journal of Engineering and Scientific Research 4(9) (2013) 1814-1822. T. K. Shinoj, J. J. Sunil, Intuitionistic fuzzy multisets and its application in medical diagnosis, Int. Journal of Mathematical and Computational Sciences 6 (2012) 34-38. T. K. Shinoj, J. J. Sunil, Intuitionistic fuzzy multisets, Int. Journal of Engineering Science and Innovative Technology 2(6) (2013) 1-24. T. K. Shinoj, J. J. Sunil, Accuracy in collaborative robotics: an intuitionistic fuzzy multiset approach, Global Journal of Science Frontier Research Mathematics and Decision Sciences, 13(10) (2013). D. Smith, M. Eggen, R.A. St. Andre, Transition to advance Mathematics, 4th Ed. New York:Brook/Cole (1997). R. R. Yager, On the theory of bags, Int. J. of General Systems 13 (1986) 23-37. L.A. Zadeh, Fuzzy sets, Inform. and Control 8 (1965) 338-353. Szmidt, E., Kacprzyk, J., On measuring distances between intuitionistic fuzzy sets, Notes on IFS 3 (4) (1997) 1-3. Szmidt, E., Kacprzyk, J., Distances between intuitionistic fuzzy Sets, Fuzzy Sets and Systems 114 (3) (2000) 505-518. P.A. Ejegwa, O.P. Edibo, Some distance measures between intuitionistic fuzzy multisets, Int. Journal of Scientific & Technology Research 3 (4) (2014) 332-334. P.A. Ejegwa, J. A. Awolola, Intuitionistic fuzzy multisets in binomial distributions, Int. Journal of Scientific & Technology Research 3 (4) (2014) 335-337.