Some Applications of -Cuts in Fuzzy Multiset Theory 1

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1 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved Some pplications of -Cuts in Fuzzy Multiset Theory 1 D Singh, 2 J lkali and 3 I Isah 1 Former professor (Indian Institute of Technology ombay), Professor, Department of Mathematics, hmadu ello University, Zaria, Nigeria 2 Research Scholar (Corresponding uthor), Department of Mathematics, hmadu ello University Zaria-Nigeria 3 Research Scholar, Department of Mathematics, hmadu ello University Zaria-Nigeria STRCT The paper briefly describes fuzzy set, multiset, fuzzy multiset, -cuts and related results In particular, it is shown how -cuts turn out to be an intertwining thread between multiset and fuzzy multisets Keywords: Fuzzy set, crisp multiset, Fuzzy Multiset, -cut 1 INTRODUCTION It is well-known that the two basic restrictions viz, the elements of a set must be definite and distinct underlay the foundation of classical set theory Classically, a set is completely characterized by a membership function restricted to { } which is also referred to as a twovalued characteristic function or simply characteristic function In other words, a set on a universe is a function { } That is, given an obect of it is decided whether or not, belongs or does not belong to However, if mathematics is required to model the concept of a class of obects and relations between them which do not possess crisply defined criteria, the classical paradigm is found incapacitated [1] For example, the classes defined by predicates involving uncertainty, ambiguity or vagueness, such as class of all real numbers much greater than one, class of all very handsome statesmen across the world, etc, cannot be adequately dealt within a standard set theoretic framework The uncertainty arises in deciding whether or not an obect of the universe belongs or does not belong to such an illdefined class Yet, the fact remains that such imprecisely defined classes play an important role in human thinking, particularly in the domain of pattern recognition, communication of information and abstraction [2] s indicated in [3], fuzziness is the rule rather than the exception in problems of pattern classification The concept in view is that of a fuzzy set 11 FUZZY SETS In view of the significant role played by settheoretic foundations, it was natural to look for a set theory-like framework to study imprecisely defined classes, particularly the one in which source of vagueness is not the presence of random variables rather the absence of precisely defined criteria for its membership function L Zadeh [2], by way of relaxing the restriction of definiteness imposed on obects to form an ordinary set, was the first to formulate such a generalized set-theoretic framework and titled it fuzzy set theory The fuzzy set theory is a mathematical theory to model vagueness and other loose concepts In order to comprehend the notion of loose concepts like classes with vague boundaries, etc, Zadeh [2] suggests to note that the notion of belonging in such cases is not the same as it has in ordinary sets Zadeh defines a fuzzy set as a generalized set of obects occurring with a continuum of degrees (grades) of membership In other words, a fuzzy set on a given nonempty universe is defined by a membership function (called generalized characteristic function, first introduced in [4]): The membership degree structure is supposed to be graded ie, it should possess a suitable ordering structure with as universal lower bound, and as universal upper bound The grade of membership (full or partial) or non-membership (full or partial) of an element in a fuzzy set corresponds to the degree to which it is compatible with the concept represented by that fuzzy set [5] n obect of the universe belongs to a fuzzy set with a higher/lower grade as the value of is nearer/farther from unity On a naïve interpretation, this means that a fuzzy set is said to differ drastically (say, for example) from a fuzzy set, whenever is obtained from by introducing a large change in membership degree of some element of the universe of discourse of Thus, a fuzzy set is defined by a continuous generalization of the set characteristic function Classes with vague defining predicates are studied by allowing partial memberships where the transition from member to member is graded s noted in [6], membership degrees form not only a superset of the standard truth degree set { }, but 328

2 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved an algebraic structure of degrees between the extremal degrees and It is the membership degree behavior of the obects of a fuzzy set that characterizes it In effect, a fuzzy set is not only characterized by but identified with a suitable membership function More perspicaciously, identifying the set and the membership function that characterize a fuzzy set provides a closer understanding of fuzzy sets The following two comments should be noted at this end: Fuzzy set is essentially non-statistical lthough the membership function of a fuzzy set does have some resemblance to a probability function when is countable (or a probability density function when is a continuum), probability theory is not found suitable to deal with the kind of uncertainty that appears especially in pattern classification and information processing In fact, this type of uncertainty is more like an ambiguity than a statistical variation ([2], [6], and [7]) Probability approach cannot deal with the kind of vagueness if its meaning refers to absence of sharp boundaries based formulations in the literature ([6] and [11], for details) Some basic definitions: Union The union of two fuzzy sets and with respective membership functions and is the fuzzy set, written as whose membership function is related to those of and by Intersection The intersection of two fuzzy sets and with respect to function as given above, is the fuzzy set, written as, whose membership function is related to those of and by Inclusion is contained in (or, equivalently, is a subset of or is smaller than or equal to ) if and only if In symbols Secondly, the notion of fuzzy sets and rough sets, though not unrelated, are two distinct approaches for modeling vagueness and uncertainty Rough sets were introduced by Z Pawlak [8] as a generalization of classical sets rough set is a crisp set whose membership function is coarsely described, sometimes called rough membership function The impreciseness of the said description arises from the lack of complete information rather than fuzziness Essentially, a rough set is defined in a universe endowed with an equivalence relation characterizing the indiscernibility relation between its obects, where as a fuzzy set is defined in an arbitrary universe devoid of such a relation ([9] and [10], for details) thorough comparative study of fuzzy sets and rough sets has been conducted in [11] In the past four decades, there have been numerous studies on the topic Most of the basics of fuzzy sets can be found in [3], [7] and [9] Goguen [7], as suggested in Zadeh [2], extended the study of fuzzy sets by taking a suitable partially ordered structure (Lattice in this case) as the range of membership function in place of an usually chosen membership algebraic structure designated by the unit interval of the real line rown [3] developed a modified version of Zadeh [2] on replacing by a oolean lattice Note, however, that most of the algebraic structures chosen as an alternative to, should have extremal elements and Seising [12] provides an excellent account of the development toward formalization of the concept of fuzziness In passing, it should be noted that, besides many formalizations of fuzzy set theory based on set-theoretic foundation, there has been a number of logic (especially, many-valued logic and model logic) 12 MULTISETS nother equally important generalization of classical set theory, by way of relaxing the restriction of distinctness on the nature of the obects forming a set was made which gave rise to multiset theory multiset (mset, for short) or a bag is an unordered collection of obects in which, unlike an ordinary set, obects are allowed to repeat Each individual occurrence of an obect in an mset is called its element ll duplicates of an obect in an mset are treated indistinguishables The obects of an mset are its distinguishable (distinct) elements The number of occurrences of an obect, which is usually finite, in an mset is called its cardinality, denoted by m (x) or C (x) or (x) The cardinality of an mset is the sum of the multiplicities of all its obects, denoted by C() or That is, C() = C (x) for all x in The root or support or carrier of an mset, denoted by *, is the set containing all distinct elements of It follows that every mset must have a unique root set The cardinality of the root set of an mset is called its dimension n mset is usually represented by using square brackets, instead of curly brackets, to distinguish it from set representations For example an mset containing one occurrence of a, two occurrences of b, and three occurrences of c is notated [a, b, b, c, c, c] or [a, b, c] 1,2,3 or [a, 2b, 3c] or [a1, b2, c3] or [1/a, 2/b, 3/c] or [a 1, b 2, c 3 ] etc For convenience, the curly brackets are often used if no confusion arises There are many other forms of multiset representations The representation of an mset in a function form, which is found quite useful especially in 329

3 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved developing axiomatic foundations and applications in computer science, is as follows: Let D be a domain set (universe) and let T be a numeric set Then, a map : D T is called a set, if T = {0, 1}; a multiset, if T = N, the set of natural numbers including zero; a signed multiset (or, hybrid/shadow set), if T = Z, the set of integers; a fuzzy (or hazy) set, if T = [0, 1] Multisets have found several applications in mathematics, computer science (especially in database theory, calculus, membrane computing, etc), linguistics, economics, etc ([13], [14], and so on) are recent references which contain most of the details regarding mathematics of multisets and their applications Some basic definitions: Union Let,, C, be multisets over a given generic set D The union of and, denoted, is the mset defined by m (x) = m (x) m (x) = maximum {m (x), m (x)}, being the union of two numbers That is, an obect z occurring a times in and b times in, occurs maximum {a, b} times in, if such maximum exists, otherwise the minimum of {a, b} is taken which always exists It follows that for any given mset X there exists an mset Y which contains elements of elements of X, where the multiplicity of an element z in Y is the maximum multiplicity of z as an element of elements of X along with the above stipulation on the existence of such a maximum We denote this fact Y = X It is clear, Dom( X) = {Dom (), X} and that the multiplicity of z in Y is the maximum multiplicities as an element of elements of X if it exists, otherwise the minimum is taken For example, let = [a, b, c, c, c], = [b, b, c, d, d] then = [a, b, b, c, c, c, d, d] hence X = X * Intersection The intersection of the msets and, denoted, is the mset C defined by m (x) = m (x) m (x) = minimum {m (x), m (x)}, being the intersection of two numbers That is, an obect x occurring a times in and b times in, occurs minimum {a, b} times in, which always exists In general, for a given mset X, Dom( X) = {Dom(): X} and z X implies that the multiplicity of z is the minimum of its multiplicities as an element of elements of X For example, let = [a, a, a, b, b, c], = [a, a, b, c, d, d] then = [a, a, b, c] The above definition can be extended to an arbitrary number of msets as follows: i = {m ii (x)x m ii (x) = min ii m i (x), for all xd} Note that for any mset X, we have X X Inclusion Let and be two multisets, is a multisubset or a submultiset of, written as or, if m (x) m (x) for all xd lso, if and, then is called a proper submset of n mset is called the parent in relation to its msubsets It is easy to see that is antisymetric ie, and =, and it is a partial ordering on the class of msets defined on a given generic domain Clearly, is a submset of every mset For example, [a, b] 1,2 is a submset of [a, b, c] 1,3,1 and the latter is a parent mset of the former submset of a given mset is called full if it contains all obects of the parent mset For example, [a, b, c] 1,2,3 is a full msubset of [a, b, c] 2,3,4 ddition The addition or sum or merge of msets and, denoted, is the mset defined by = m (x) + m (x), for any x D, being the arithmetic addition of two msets That is, an obect x occurring a times in and b times in, occurs a + b times in This, being the direct sum or arithmetic addition of two numbers, is sometimes called the counting law For example, let = [a, a, a, b, b, c], = [a, a, b, c, d, d] then = [a, a, a, a, a, b, b, b, c, c, d, d] Moreover, for a finite mset X, the maximum multiplicity of elements of elements of X always exists However, for certain infinite sets such as X = {{y}, [y] 2, [y] 3, } the maximum multiplicity of elements of elements of X does not exist, and hence X = {y} y the definition of union it is obvious that the multiplicity of any y X is irrelevant to X, 13 FUZZY MULTISETS In view of the aforesaid two very dominant generalizations of the classical set theory, efforts to develop mathematical structures characterizing classes of more complex obects, possessing both fuzziness and multiplicity, have been made The concept in view is that of a fuzzy multiset In order to keep a parity, a non-fuzzy 330

4 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved multiset is called a crisp multiset Fuzzy sets were introduced in yager [15] Formally, a fuzzy multiset in some universe set is a crisp multiset of That is, a fuzzy multiset is a multiset of pairs, where the first part of each pair is an element of X and the second part is the degree to which the first part belongs to that fuzzy multiset (see [16] for details) For example, { }, equivalently written as, {{ } { } }}, is a fuzzy multiset in { } Note that an element of may occur more than once with possibly the same or different membership values For example, two with the same multiplicity belong to the fuzzy multiset { } of X { x, y, z, w}, equivalently written as {{ } { } }} Thus, count is a finite multiset of the unit interval For, the membership sequence is defined as a (monotonic) decreasingly sequence of the elements of count, denoted by where 1 2 p ( ( ), ( ),, ( )) 1 2 ( ) ( ) p x x ( x) x x x, Note that for defining an operation between two fuzzy multisets and, the lengths of the membership sequences ( x), ( x),, ( x ) and 1 2 p ' 1 2 p ( x), ( x),, ( x) need to be set equal Lengths L( x; ) and are respectively defined as L( x; ) max{ : ( x) 0} ; and L( x;, ) max{ L( x; ), L( x; )} For brevity, for or is also used if no confusion arises Some basic fuzzy multisets operations are defined as follows: Union ( x) ( x) ( x), 1,, L( x), x X For the example taken above, {{07,03}/ x,{1,07,06}/ y,{08,02}/ w} Intersection ( x) ( x) ( x), 1,, L( x), x X For the example taken above, {{04,0}/ x,{09,06,0}/ y,{0,0}/ w} or simply {{04}/ x,{09,06,}/ y} Inclusion ( x) ( x), 1,, L( x), x X Thus, and ddition is defined by the addition operation in as defined for crisp multisets That is, for fuzzy multisets = {(x i, µ i),, (x k, µ k)} and = {(x p, µ p),, (x r, µ r )}, we have = {(x, µ ),, (x, µ ), (x, µ ),, (x, µ )} i i k k p p r r For example, given {{04, 03}/x, {1, 06, 06}/y,{0,0}/w } and = {{07,0}/x, {09, 07,0}/y, {08, 02}/w}, = {{07,04,03}/x, {1,09,07,06,06)/y,{08,02}/w} For example, fuzzy msets {{04, 03}/x, {1, 06, 06}/y} and = {{07}/x, {09, 07}/y, {08, 02}/w} need to be arranged as {{04, 03}/x, {1, 06, 06}/y,{0,0}/w} and = {{07,0}/x, {09, 07,0}/y, {08, 02}/w} Here,,,, See [2], [3], [11], [17], [18], and [19], for details 2 - CUTS (OR - LEVEL MSETS) ND SOME RELTED RESULTS In this section, we define - cuts and outline some of their properties It is to be noted that the definition of - cuts for fuzzy multiset is a generalization of that for fuzzy sets 331

5 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved 21 INTRODUCTION The application of α-cuts or α-level sets was made in Zadeh [2] with the aim of establishing a bridge between fuzzy set theory and classical set theory s mentioned above, a fuzzy set is a collection of obects with various degrees of membership It is often useful to consider those elements that have at least some minimal degree of membership, say α This is something like asking who has a passing grade in a class or a minimal height to ride on a roller coaster This process can be better comprehended by using the notion of alpha-cuts Many scholars have used the concept in fuzzy set theory [20, 21, 22] Klir and Yuan [20] show that all alphacuts and all strong alpha-cuts of any fuzzy set form two distinct families of nested crisp sets, and that a fuzzy set is convex if any point in between two other points resides in the alpha-cut that is largest Sun and Han [21] provide a series of α- related and interval related concepts They introduce the notion of inverse α-cut to improve the usability to solve some of the concept of α-cuts to real life problems, which they show can also be considered as a bridge between fuzzy sets and crisp sets It is demonstrated that, like the α-cuts, every fuzzy set can uniquely be represented by the family of all its inverse α-cuts This representation allows us to extend various properties of crisp sets and operations on crisp sets to their fuzzy counterparts Chutia et al [23] and Mahanta et al [24] claimed that, the standard method of -cuts fails to find the square root of a fuzzy number, hence, does not always yield results s such they provide an alternative method which they claim can be utilized in the cases where the method of -cuts fails Dutta et al [22] refute their claim by providing an alpha-cut method for finding n th root of a fuzzy number, which is even simpler than their proposed method In addition, they provide methods for performing different type of fuzzy arithmetic operations including exponentiation and taking log Kreinovich [25] observes that membership functions and α-cuts could be viewed as two alternative representations of a fuzzy number; as each imprecise (fuzzy) property can be described by a membership function ie, by a mapping μ from real numbers into the interval [0,1] lternatively, this same property can be described by α-cuts ie, by a function that maps each number α from the interval [0, 1] into an interval x(α) = {x : μ(x) α} It is shown that from the traditional mathematical viewpoint, these two representations are equivalent, however, the two approaches are not algorithmically equivalent In particular, from application point of view it is observed that only α-cuts representation guarantees algorithmic fuzzy data processing and therefore enables us to efficiently process fuzzy data 22 DEFINITIONS ND RESULTS The α-cut (α (0, 1]) for a fuzzy multiset, denoted, is a crisp multiset defined as follows: ( x) count ( x) 0 1 ( x), ( x) count ( x), 1,, L( x) The strong α-cut (α [0, 1]), denoted ][ α, is a crisp multiset defined as follows: ( x) count ( x) 0 1 ( x), ( x) count ( x), 1,, L( x) s remarked in [18], the way - cuts are defined, it is immediate to see that they represent closed (open) sets if is a metric space, is a subset of, and the membership function is continuous Some basic properties of - cuts Let,, FM(X), the class of all finite fuzzy multisets of The following results hold for all,, [0,1] : i ii implies iii iv v and and and and Proofs of (i) and (ii) follow from the definition of - cuts Proof of (iii) Part 1 Proof of For x, we have ( x ), 1,, L ( x ) m a x x[ ( x ), ( ) L] x, 1, ( x) or ( x), 1,, L( x) x or x 332

6 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved lso, x x or x ( x) or ( x), 1,, L( x) max[ ( x), ( x)], 1,, L( x) x ( x ), 1,, L ( x ) Hence, Part 2 Proof of For any x, we have ( x ), 1,, L ( x ) min[ ( x), ( x)], 1,, L( x) () x and ( x ), 1,, L ( x ) x lso x x and x () x and ( x )], 1,, L ( x ) min[ ( x), ( x)], 1,, L( x) ( x ), 1,, L ( x ) x Hence, Following the aforesaid argument proofs of (iv) and (v) can be given It is worth noting that - cuts commute with respect to set operations and From (ii) above, it is immediate to see that the mset sequences [0,1] and cuts and strong cuts [0,1] for, respectively, being monotonic decreasing with respect to, each form a nested family of crisp msets From (iii) and (iv); it follows that the standard fuzzy multiset intersection and fuzzy multiset union are cut worthy when applied to two fuzzy multisets and, due to the associativity of min and max, it also holds for a finite number of fuzzy multisets (see [3], [15], [17], [18], and [19] for many related details) Summarily, in order to have a crisp account of applications of - cuts in fuzzy multisets context, it should be noted that there exist a multitude of mathematical concepts, such as boundedness, convexity, connectedness, arc-wise connectedness, star-shaped, having holes, etc, defined in terms of - cuts ([2], [17] and [26], for details) In particular, it is immediate to see that the collection of fuzzy multisets, akin to that of ordinary fuzzy sets on a universe, forms a distributive lattice though not a complemented one ([26], for details) Note that, the central idea underlying most of the results presented above have precedents in terms of crisp sets and fuzzy sets It is well-known that a fuzzy set can be related to a family of crisp sets through the notions of - cuts or - level sets 3 DECOMPOSITION THEOREMS FOR FUZZY MULTISETS In this section, it is shown how - cuts help relate crisp multisets and fuzzy multisets (see [2] and [18], for details) We consider a simple crisp multiset to motivate our discussions Let {02,03,05}/ x,{05}/ y,{01,08,09}/ z be a fuzzy multiset We show below how can be represented by crisp multisets using - cuts Following our elaboration above, it can be seen that the given fuzzy multiset is associated with six distinct - cuts, defined by the following characteristic functions viewed as special membership functions: {1,1,1}/ x,{1}/ y,{1,1,1}/ z, x y z 03 x y z 05 x y z 08 x y z {0,0,0}/ x, 09 We now convert each of the cuts multiset follow: {1,1,1}/,{1}/,{0,1,1}/, 02 {0,1,1}/,{1}/,{0,1,1}/, 03 {0,0,1}/,{1}/,{0,1,1}/, 05 {0,0,0}/,{0}/,{0,1,1}/, 08 {0}/ y,{0,0,1}/ z, 09 to a special fuzzy, defined for each xx [ x, y, z], as (31) Thus, we obtain {{01,01,01}/ x,{01}/ y,{01,01,01}/ z}; {{02,02,02}/ x,{02}/ y,{0,02,02}/ z}; {{0,03,03}/ x,{03}/ y,{0,03,03}/ z}; {(0,0,05) / x,(05) / y,(0,05,05) / z}; {{0,0,0}/ x,{0}/ y,{0,08,08}/ z}; {{0,0,0}/ x,{0}/ y,{0,0,09}/ z} 333

7 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved It is now easy to see that the standard fuzzy multiset union of these six special fuzzy multisets is exactly the original fuzzy multiset That is where fuzzy multiset union are as defined in (31) and is the standard The aforesaid representation of an arbitrary fuzzy multiset in terms of fuzzy multiset the decomposition of, defined in terms of cuts of, is usually referred to as a Similarly, following the reference fuzzy multiset can be represented in terms of, a defined in terms of the strong - cuts In the following, we describe two decomposition theorems for fuzzy multisets in terms of -cuts and strong -cuts, respectively Theorem 31 First Decomposition Theorem For every FM ( X ), [0,1] where fuzzy multiset union are as defined in (31) and (32) is the standard Proof For each particular x X, let b ( x), 1,, L( x) Now, for each ( b,1], we have ( x ) b, 1,, L ( x ) and, therefore, 0 On the other hand, for each (0, b], we have ( x ) b, 1,, L ( x ) Therefore, Hence, ( x) Sup b ( x), 1,, L( x) [0,1] (0, b] Since the same argument is valid for each x X, it follows that each fuzzy multiset can be uniquely represented by the family of all its -cuts Theorem 32 Second Decomposition Theorem For every FM ( X ), [0,1] (33) Proof The proof is analogous to the proof of theorem 31 For each particular x X, let b ( x), 1,, L( x) then [0,1] [0,1] ( x) Sup (0,1] max Sup, Sup (0, b) ( b,1] That is, ( x) Sup b ( x), 1,, L( x) (0, b) Hence, a fuzzy multiset can also be uniquely represented by the family of all its strong -cuts 4 CONCLUDING REMRKS It has been shown that either of the aforesaid representations allows us to extend various properties of crisp multisets and operations on crisp multisets to their fuzzy multiset counterparts These extensions are cut worthy since in each representation, a given crisp property or operation is required to be valid for each crisp multiset involved in the representation s indicated in [19], application of the concepts from nonstandard analysis could be of interest in studying Fuzzy multisets [for example, representing a half open fuzzy point as a closed interval where is a positive infinitesimal number: for all real number ] pparently, the full potential of - cuts is yet to be exploited REFERENCES [1] D Singh, On Cantor s Concept of a Set, International Logic Review, 32 (1985) [2] L Zadeh, Fuzzy Sets, Information Control, 8 (1965) [3] J G rown, note on Fuzzy sets, Information and Control, 18 (1971) [4] H Whitney, Characteristic Functions and The lgebra of Logic, nnals of Mathematics, 34 (1933)

8 Vol 5, No 4 pril 2014 ISSN CIS Journal ll rights reserved [5] Kasko, Fuzzy Thinking: The New Science of Fuzzy Logic, Hyperion press, (1993) [6] S Gattwald, Universes of Fuzzy Sets and xiomatization (part 1), Studia Logica: n International J Symbolic Logic, 82 (2) (2006) [7] J Goguen, L-fuzzy Sets, Journal of Mathematical nalysis and pplications, 18 (1967) [8] Z Pawlak, Rough Sets, International J Inform Compt Sci, 11 (5) (1982) [9] M Wygralak, Rough Sets and Fuzzy Sets Some Remarks on Interrelations, Fuzzy Sets and Systems, 29 (1989) [10] M Wygralak, Further remarks on the relation between rough and fuzzy sets, Fuzzy Sets and Systems, 47 (1992) [11] Y Y Yao, comparative study of fuzzy sets and Rough sets, Information Sciences, 100 (1 4) (1998) [12] R Seising, Vagueness, Haziness, and Fuzziness in Logic, Science and Medicine-efore and when Fuzzy Logic egan, On Web (ISCSE), 2005 [13] D Singh, M Ibrahim, Y Tella, and J N Singh, n Overview of pplications of Multisets, Novi Sad Journal of Mathematics, 37 (2) (2007) [14] W lizard, The Development of Multiset Theory, Modern Logic, 1 (1991) [15] R R Yager, On the Theory of ags, International Journal of General Systems, 13 (1986) [16] D Singh, lkali, and M Ibrahim, n Outline of the Development of the Concept of Fuzzy Multisets, International Journal of Innovation, Management and Technology, 4 (2) (2013) [17] S Miyamoto, Fuzzy multisets with infinite collections of memberships, Proc of the 7 th International Fuzzy System ssociation World Congress (IFS 97), (1997) [18] S Miyamoto, Fuzzy Multisets and Their Generalizations: in Claude et al (Eds) Multiset Processing: Mathematical, Computer Science and Molecular Points of View LNCS, 2235, Springer, (2001) [19] S Miyamoto, Remarks on basics of fuzzy sets and fuzzy multisets, Fuzzy Sets and Systems, 156 (2005) [20] G J Klir and Yuan, Fuzzy Sets and Fuzzy Logic: Theory and pplications, Prentice Hall, NJ (1995) [21] Z Sun and J Han, Inverse alpha-cuts and Interval [a, b)-cuts, Proceedings of the International Conference on Innovative Computing, Information and Control (ICICIC2006), 30 ugust - 1 September, eiing, IEEE Press, (2006) [22] P Dutta, H oruah and T li, Fuzzy rithmetic with and without using α-cut method: Comparative Study, International Journal of Latest Trends in Computing (E- ISSN: ) 2 (1) (2011) [23] R Chutia, S Mahanta, and H K aruah, n alternative method of finding the membership of a fuzzy number, International ournal of latest trends in computing, 1 (2) (2010) [24] S Mahanta, R Chutia, and H K aruah, Fuzzy rithmetic without using the method of α- cut, International ournal of latest trends in computing, 1 (2) (2010) [25] V Kreinovich, Membership Functions or α-cuts? lgorithmic (Constructivist) nalysis Justifies an Interval pproach, pplied Mathematical Sciences, 7 (5) (2013) [26] E W Chapin, Jr, Set-valued set theory: part one, NDJFL, 15 (4) (1974) UTHORS PROFILE Singh, D is a former professor (IIT ombey) and currently professor, Department of Mathematics, hmadu ello University, Zaria, Nigeria reas of specialization are Sets, Multisets, Fuzzy set, Fuzzy multisets and soft sets theory lkali, J, received his M sc degree in Mathematics from hmadu ello University, Zaria-Nigeria in 2002 He is currently, a PhD research scholar and lecturer in the department of Mathematics, hmadu ello University, Zaria-Nigeria Isah, I received his M sc degree in Mathematics from hmadu ello University, Zaria-Nigeria in 2012 He is currently, a PhD research scholar and lecturer in the department of Mathematics, hmadu ello University, Zaria-Nigeria 335