Fuzzy set operations

Let A and B be two fuzzy sets within a universe of diseourse U with membership funetions /ta and /tb respeetively. The following fuzzy set operations ean be defined as... 

Accordingly, the breadth-first check procedure is carried out by using a series of membership function values that weight each extension and allow selection of the most likely and discard the less likely extensions. Up to now the selection has been carried out by the user, as depicted in Fig. 13, but with the further development of more reliable membership functions, the fuzzy set operations discussed in the previous section may be applied to render the CASE process fully automatic. 

Fuzzy set operations are derived from classical set theory. In addition, there exist theories for calculating with fuzzy numbers, functions, relations, measures, or integrals. 

Fuzzy set operations are a generalization of crisp set operations, each of which is a fuzzy set operation. Infuzzy logic, three operations, including fuzzy complement, fuzzy intersection and fuzzy union, are the most commonly used. Let fuzzy sets ... 

Fuzzy Encoding and Fuzzy Set Operations in Genetic Algoritms... 

Figure 10.38 shows an input window with three triangular fuzzy sets NB, Z and PB. Each set is positioned in its regime of operation by the centre parameter c so that, for example, NB can only operate on the negative side of the universe of discourse. The width of each set is controlled by parameter ri . 

Ravi, V. and Reddy, P.J. (1998) Fuzzy linear fractional goal programming applied to refinery operations planning. Fuzzy Sets and Systems, 96, 173. 

Morsi, N. N. (1994) Hyperspace fuzzy binary relations Fuzzy Sets and Systems. 67(2) 221—37. [The author associates with each implication operator in (0, l)-valued logic, under certain conditions, an algorithm for extending a fuzzy or ordinary binary relation psi from X to Y, to a fuzzy binary relation from 1(20 to 1(f) said to be a fuzzy hyperspace extension of psi.]... 

Fuzzy control, which is based on fuzzy sets theory proposed by Zadeh [9], can easily utilize empirical knowledge gained from skilled operators by employing... 

The final degree-of-satisfaction resulting from certain variable set, nv(xy) can be determined by aggregating the degree-of-satisfaction for all objectives, /ijs (xv), s e S. via specific fuzzy intersection operator, T. 

The fundamental properties for a fuzzy set and the related operators can be found in [15], As the firing level for each policy is determined by the above procedure, the best solution, x, with the maximal firing level, nv(Xy), can be selected. 

However, the identification of the fuzziness associated with single parameter characterizing foodstuffs is not enough. The authentication process is not usually restricted to a single parameter but in fact there are often several of them. If we want to operate with their membership function (e.g. low linoleic and high 24-methylene cycloarthanol), we need to define operations on the fuzzy set. Thus, the classical rule R = IF (input) A, THEN (output) B can be extended... 

Any property or operation extended from classical set theory into the domain of fuzzy set theory that is preserved in all a-cuts is called a cutworthy property or operation-, if it is preserved in all strong a-cuts, it is called a strong cutworthy property or operation. It is important to realize that only some properties and operations involving fuzzy sets are cutworthy or strong cutworthy. They are of special significance since they bridge fuzzy set theory with classical set theory. They are like reference points from which other fuzzy properties or operations deviate to various degrees. 

Contrary to their classical counterparts, operations on fuzzy sets are not unique. This is a natural consequence of the fact—well-established by numerous psychological experiments—that logical connectives (not, and, or, etc.) in linguistic expressions have different meanings when applied by... 

Fuzzy complements, intersections, and unions have been characterized and studied on axiomatic grounds. Efficient procedures are now available by which various classes of functions can be generated, each of which covers the whole recognized semantic range of the respective operation. In addition, averaging operations for fuzzy sets, which have no counterparts for crisp sets, have also been investigated in this way. This rather theoretical subject, which is beyond the scope of this overview, is thoroughly covered in ref. 18. 

Fuzzy sets on [R that satisfy these requirements capture various linguistic expressions, describing approximate numbers or intervals, such as numbers that are close to a given real number or numbers that are around a given interval of real numbers. Moreover, we can define meaningful arithmetical operations on these fuzzy sets via the a-cut representation. At each a-cut, these operations are the standard arithmetical operations on closed intervals ... 

When fuzzy sets are defined on universal sets that are Cartesian products of two or more sets, they are called fuzzy relations. For any Cartesian product of n sets, the relations are called n-dimensional. From the standpoint of fuzzy relations, ordinary fuzzy sets may be viewed as degenerate, one-dimensional relations. All concepts and operations applicable to fuzzy sets are applicable to fuzzy relations as well. However, fuzzy relations involve additional concepts and operations that emerge from their multidimensionality. 

All of these linguistic terms except fuzzy modifiers are represented in each context by appropriate fuzzy sets. Fuzzy predicates are represented by fuzzy sets defined on universal sets of elements to which the predicates apply. Fuzzy truth values and fuzzy probabilities are represented by fuzzy sets defined on the unit interval [0,1]. Fuzzy quantifiers are either absolute or relative they are represented by appropriate fuzzy numbers defined either on the set of natural numbers or on the interval [0,1]. Fuzzy modifiers are operations by which fuzzy sets representing the various other linguistic terms are appropriately modified to capture the meaning of the modified linguistic terms. 

As is well known, fuzzy sets are holistic concepts, each representing a potentially infinite family of nested crisp sets (a-cuts), each defined for a particular number (a) in the unit interval. When we operate on fuzzy sets, we operate in fact simultaneously on all crisp sets in the associated families. This is quite powerful, both conceptually and computationally. 

For fuzzy sets used in this study various set operations are required. If A and B are fuzzy subsets of U, then the fuzzy intersection, that is, the result of the operation A and B is denoted by A i B, and is interpreted as a fuzzy subset C of set U, where the corresponding membership... 

A fuzzy set A is said to have the symmetry element R corresponding to the symmetry operator R if and only if for every point x and for the transformed point... 

A fuzzy symmetry operator R(ij) of fuzzy symmetry element R()3 ) present for fuzzy set A at the fuzzy level j3 of the fuzzy Hausdorff-type... 

In general, the elements of a fuzzy set A are fully specified if their membership functions are given, that is, if for the elements x of the underlying space X, the pairs (jt, /a (x)) are specified. Using this pair notation, the action of operator r can be described by the defining equation... 

Note that if a fuzzy set A has the fuzzy symmetry element / ()3) corresponding to the symmetry operator R at the fuzzy level j8 of the fuzzy Hausdorff-type similarity measure then the application of R on 4 generates a set R indistinguishable from set A at the fuzzy level (i. For the fuzzy set A the application of symmetry operator R of fuzzy symmetry element R(p ) present at the fuzzy level is completed by a formal recognition of the indistinguishability of set R/1 and set A at the given fuzzy level. This additional step, for which the sufficient and necessary condition is the presence of fuzzy symmetry element Ri ) at the fuzzy level /S, involves operator setting the membership functions of elements of the fuzzy set R/1 equal to those of fuzzy set A. 

A fuzzy set A has the / fuzzy symmetry group G Sg, /3 ) at fuzzy level if has the fuzzy symmetry element R( j8 ) at the fuzzy level j8 of the fuzzy Hausdorff similarity measure for each symmetry operation R of the crisp symmetry group G. The )3 fuzzy symmetry group G(s, at the fuzzy level /3 that is the supremum of the levels j8 of all )3 fuzzy symmetry groups G(Sg, (3 ) of A is the fuzzy symmetry group G s, of the fuzzy set A ... 

This family of operators can be regarded as an extension of the family of point symmetry operators. Symmorphy is a particular extension of the point symmetry group concept of finite point sets, such as a collection of atomic nuclei, to the symmorphy group concept of a complete algebraic shape characterization of continua, such as the three-dimensional electron density cloud of a molecule. In fact, this extension can be generalized for fuzzy sets. 

By analogy with fuzzy symmetry elements, a fuzzy set A is said to have the fuzzy symmorphy element S( (3) corresponding to the symmorphy operation S at the fuzzy level /3 of the fuzzy Hausdorff-type similarity measure Sg if and only if the fuzzy similarity measure Sg between Sv4 and A is greater than or equal to /3 ... 

In the limiting case of y3 = 1, similarity becomes indistinguishability. In this case, the fuzzy symmorphy element S ) becomes an ordinary symmorphy element corresponding to the symmorphy operation S. The maximum fuzzy level l3(A,S,Sg) at which the fuzzy symmorphy element S( j8) is present for the fuzzy set A is given as... 

A Juzzy symmorphy operator S( g) of fuzzy symmorphy element 5( 0 exhibited by fuzzy set A at the fuzzy level )8 (according to the fuzzy Hausdorff-type similarity measure Sg) is defined by its action on the given fuzzy set A ... 

Operator has the same role as operator in the case of fuzzy symmetry operators. To exploit the fuzzy indistinguishability of sets A and Szl at the fuzzy level )3, operator resets the values of fuzzy membership functions if and only if the fuzzy symmorphy element... 

By analogy with fuzzy symmetry operations, if a fuzzy set A has the three fuzzy symmorphy elements 5( /3), 5 ( /3) and S"( ) at fuzzy level and if the product... 

---