IJMMS, Vol. 8, No. 1, (June 2012) : 103-107 Serials Publications ISSN: 0973-3329 SOME OPERTIONS ON INTUITIONISTIC FUZZY SETS Hakimuddin Khan bstract In This paper, uthor Discuss about some operations on Intuitionistic Fuzzy sets (or vague sets) we understand that the most important and potential generalization of Fuzzy sets came in the form of intuitionistic Fuzzy sets developed by tnassove. It is justified by Bustince and Burillo[2] that intuitionistic fuzzy sets and vague sets are same. So in this paper the two terminologies vague sets and intuitionistic fuzzy sets have been used with same meaning and objectives. The actual contribution in this paper is to discuss some operations on intuitionistic fuzzy sets or vague sets. 1. INTRODUCTION We assumes that the parameters of a model represent exactly either our perception of the phenomenon modeled or the features of the real system that has been modeled. Generally we indicates that the model is unequivocal, that is, it contains no ambiguities. By crisp we mean yes-or-no type rather than more-or-less type. In conventional dual logic, for instance, a statement can be true or false-and definitely nothing in between. Vagueness, and uncertainty have so far been modeled by classical set-theoretic approach. ccording to this approach, borderline elements can be either put into the set or should be kept outside it. Hence it becomes inadequate for applying to humanistic problems. Zadeh [12], in 1965 initiated the notion of fuzzy set theory as a modification of the ordinary set theory, which plays a very important role to solve some real life problems. Vague sets can be modeled using this theory. fuzzy set is a class of objects in which the transition form membership to non-membership is gradual rather than abrupt. Such a class is defined by a membership function which assigns to an element a grade or degree of membership between 0 and 1. For a beginner on fuzzy set theory, the work in [3], [7], [8], [9], [11], [13], etc. are good enough to start with. It may be accepted that the results of the study of a mathematical system will be valid for each of those otherwise different situations which provided motivation and inspiration for the same. Such a study also provided an economy of effort and leads to a better and fuller understanding of the motivation situations.
104 Hakimuddin Khan Even without considering the motivation situations inherent in cybernetics and general systems prevailing in the emerging man- machine civilization. Different authors from time to time have made a number of generalizations of fuzzy set theory of Zadeh [12] with different objectives ([2], [3], [4], [5], [6], [10]) of these, the notion of intuitionistic fuzzy set theory (IFS theory) introduced by tanassov [1] is of interest to us. ll fuzzy sets can be viewed as intuitionistic fuzzy sets. Many authors have asserted that there are a large number of real life problems for which IFS theory is a more suitable tool than fuzzy set theory. If and B are two intuitionistic fuzzy subsets of the set E, then B iff x E, [ (x) B (x) and (x) B (x)]. B iff B. = B iff x E, [ (x) = B (x) and (x) = B (x)]. = { x, (x), I B = { x, min ( (x), (x)), max ( (x), (x)) x E}. B B B = { x, max ( (x), (x)), min ( (x), (x)) x E}. B B + B = { x, (x) + B (x) B (x) (x), (x) B. B = { x, (x) (x), (x) + (x) (x) B B B? = { x, 1 (x),? = { x, 1 (x), C () = { x, K, L x E} and I () = { x, k, l x E}, where K sup ( x), L inf ( x) x E x E k inf ( x) and l sup ( x) x E x E 2. VGUE SETS RE INTUITIONISTIC FUZZY SETS Recently Gau and Buehrer reported in IEEE [5] the theory of vague sets. We have already mentioned earlier that vague sets and intuitionistic fuzzy sets are same concepts as clearly justified by Bustince and Burillo in [2]. Consequently, in this thesis the two terminologies vague set and intuitionistic fuzzy set have been used with same meaning and objectives. Fuzzy set is defined as the set of ordered pairs = {(u, (u)) : u U }, where (u) is the grade of membership of element u in set. The greater (u), the greater is the truth of the statement that the element u belongs to the set.
Some Operations on Intuitionistic Fuzzy Sets 105 But Gau and Buehrer [5] pointed out that this single value combines the evidence for u and the evidence against u. It does not indicate the evidence for u and the evidence against u, and it does not also indicate how much there is of each. The same is the philosophy with which tanassov [1] originally defined intuitionistic fuzzy sets earlier. For the sake of completeness, we present below the notion of vague sets too modeled by Gau and Buehrer [5]. Definition 1.1: Vague Set vague set (or in short VS) in the universe of discourse U is characterized by two membership functions given by: (i) (ii) a truth membership function t : U? [0, 1], and a false membership function f : U? [0, 1] where t (u) is a lower bound of the grade of membership of u derived from the evidence for u, and f (u) is a lower bound on the negation of u derived from the evidence against u, and t (u) + f (u) = 1. Thus the grade of membership of u in the vague set is bounded by a subinterval [t (u), 1 f (u)] of [0, 1]. This indicates that if the actual grade of membership is µ (u), then t (u) = µ (u) = 1 f (u). The vague set is written as = {< u, [t (u), f (u)] > : u U }, where the interval [t (u), 1 f (u)] is called the vague value of u in and is denoted by V (u). For example, consider an universe U = {DOG, CT, RT}. vague set of U could be = {< DOG, [.7,.2] >, < CT, [.3,.5]., < RT, [.4,.6] >}. It is worth to mention here that interval-valued fuzzy sets, are not vague sets. In i v fuzzy sets, an interval valued membership value is assigned to each element of the universe considering the evidence for u only, without considering evidence against u. In vague sets both are independently proposed by the decision maker. This makes a major difference in the judgment about the grade of membership. Note: Intuitionistic Fuzzy Sets (or Vague Sets) have an extra edge over fuzzy sets. There are a number of generalizations of fuzzy sets of Zadeh done by different authors. For each generalization, one (or more) extra dimension is annexed with a more specialized type of aims and objectives. Thus, a number of higher order fuzzy sets are now in literatures and are being applied into the corresponding more specialized application domains.
106 Hakimuddin Khan While fuzzy sets are applicable to each of such application domains, higher order fuzzy sets can not, because of its specialization character by birth. pplication of higher order fuzzy sets makes the solution-procedure more complex, but if the complexity on computation-time, computation-volume or memory-space are not the matter of concern then a better results could be achieved. Intuitionistic fuzzy sets of tanassov have also an extra edge over fuzzy sets. Let U be a universe, the set of all students of Calcutta School. Let be an intuitionistic fuzzy set of all good-in-maths students of the universe U, and B be a fuzzy set of all good-in-maths students of U. Suppose that an intellectual school-principal M 1 proposes the membership value µ B (x) for the element x in the fuzzy set B by his best intellectual capability. On the contrary, another intellectual Principal M 2 proposes independently two membership values t (x) and f (x) for the same element in the vague set by his best intellectual capability. The amount t (x) is the true-membership value of x and f (x) is the false-membership value of x in the vague set. Both M 1 and M 2 being human agents have their limitation of perception, judgment, processing-capability with real life complex situations. In the case of fuzzy set B, the Principal M 1 proposes the membership value µ B (x) and proceed to his next computation. There is no higher order check for this membership value in general. In the later case, the Principal M 2 proposes independently the membership values t (x) and f (x), and makes a check at this base-point itself by exploiting the constraint t (x) + f (x) = 1. If it is not honored, the manager has a scope to rethink, to reshuffle his judgment procedure either on evidence against or on evidence for or on both. The two membership values are proposed independently, but they are mathematically not independent. This is the breaking philosophy of intuitionistic fuzzy sets (Gau and Buehrer s vague sets [5]). Since vague sets and intuitionistic fuzzy setsare same concepts as clearly justified by Bustince and Burillo in [2], in the subsequent part of this thesis, in all definitions as well as characterizations, the two terminologies vague set and intuitionistic fuzzy set have been used with same meaning and objectives. There is no confusion in this issue. Definition 1.2: n intuitionistic fuzzy set (or a vague set) of a set U with t (u) = 0 and f (u) = 1. u U is called the zero intuitionistic fuzzy set (or zero vague set) of U. Definition 1.3: n intuitionistic fuzzy set (or a vague set) of a set U with t (u) = 1 and f (u) = 0. u U is called the unit intuitionistic fuzzy set (or unit vague set) of U. REFERENCES [1] tanassov K., (1994), New Operations Defined Over Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 61,137-142. [2] Bustince H., and Burillo P., (1996), Vague Sets are Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 79, 403-405.
Some Operations on Intuitionistic Fuzzy Sets 107 [3] Dubois D., and Prade H., (1980), Fuzzy Sets and Systems, Theory and pplications, cademic Press, London, (1980). [4] Dubois D., and Prade H., (1987), Twofold Fuzzy Sets and Rough Sets Some Issues in Knowledge Representation, Fuzzy Sets and Systems, 23, 3-18. [5] Gau W. L., and Buehrer D. J., (1993), Vague Sets, IEEE Trans. Systems Man. Cybernet., 23(2), 610-614. [6] Goguen J.., (1967), L-Fuzzy Sets, J. Math. nal. ppl., 18, 145-174. [7] Klir G. J., and Yuan B., (1995), Fuzzy Sets and Fuzzy Logic: Theory and pplications, Englewood Clifis, NJ, Prentice-Hall. [8] Lin T. Y., (1994), Fuzzy Reasoning and Rough Sets, In Rough Sets, Fuzzy Sets and Knowledge Discovery, W. P. Ziarko, (Ed.), Springer-Verlag, London, 343-348. [9] Lin T. Y., and Liu Q., (1994), Rough pproximate Operators: xiomatic Rough Set Theory, In Rough Sets, Fuzzy Sets and Knowledge Discovery, W. P. Ziarko, (Ed.), Springer-Verlag, London, 256-260. [10] Lin T. Y., (1996), Set Theory for Soft Computing, Unified View of Fuzzy Sets via Neighborhoods, Proceedings of 1996 IEEE International Conference on Fuzzy Systems, New Orleans, Louisiana, (September 8-11, 1996), 1140-1146. [11] Yazici., George R., Buckles B. P., and Petry F. E., Survey of [12] Zadeh L.., (1965), Fuzzy Sets, Infor. and Control, 8, 338-353. [13] Zadeh L.., Fu K. S., Tanaka K., and Shimura M., (Eds.), (1975), Fuzzy Sets and Their pplications to Cognitive and Decision Processes, cademic Press, New York. [14] tanassov K., (1989), More on Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 33, 37-46. [15] tanassov K., (1995), Remarks on the Intuitionistic Fuzzy Sets-III, Fuzzy Sets and Systems, 75, 401-402. [16] tanassov K., (1990), Remarks on a Temporal Intuitionistic Fuzzy Logic, 2 nd Scientific Session of Mathematical Foundation rtificial Intelligence, Sofia IM-MFIS. [17] tanassov K., (1994), Operators Over Interval Valued Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 64, 159-174. [18] tanassov K., and Gargov G., (1994), Intuitionistic Fuzzy Logic Operators of a Set Theoretical Type, FUBEST 94, Sofia, Bulgeria, 39-43, (September). [19] tanassov K., and Gargov G., (1990), Intuitionistic Fuzzy Logic, C. R. cad. Bulgaria Sc., 43(3), 9-12. [20] tanassov K., and Gargov G., (1993), New Operations Defined Over the Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, 61(2), 131-141. Hakimuddin Khan ssociate Professor, Jagannath International Management School, Vasant Kunj, New Delhi-110070, India.