Crisp sets

Fuzzy sets and fuzzy logic. Fuzzy sets differ from the normal crisp sets in the fact that their elements have partial membership (represented by a value between 0 an 1) in the set. Fuzzy logic differs from the binary logic by the fact that the truth values are represented by fuzzy sets. 

A conventional set which contains members that satisfy such precise properties concerning theii set membership is called a crisp set. 

An important property of a fuzzy set is its cardinality. While for crisp sets the cardinality is simply the number of elements in a set, the cardinality of a fuzzy set A, CardA, gives the sum of the values of the membership function of A, as in Eq. (9). 

Conventional set theory distinguishes between those elements that are members of a set and those that are not, there being very clear, or crisp boundaries. Figure 10.2 shows the crisp set medium temperature . Temperatures between 20 and 30 °C he within the crisp set, and have a membership value of one. 

The degree of membership or the membership value of an object in a set measures the extent to which the object belongs in that set. For crisp sets, the only possible membership values are 0 and 1 (Figure 8.1). A membership of 1 tells us that the object is a member of that set alone, while a membership of 0 shows that it has no membership in the set. 

Thus, propanol, C3H70H, has a membership of 1 in the three-carbon molecule class, while ethanol, C2H5OH, has a membership of 0 in the same class. As the membership in a crisp set must take one of only two possible values, Boolean (two-valued) logic can be used to manipulate crisp sets. If all the knowledge that we have can be described by placing objects in sets that are separated by crisp divisions, the sort of rule-based approach to the development of an expert system described in the previous chapter is appropriate. 

A phrase such as "seriously contaminate" is known as a linguistic variable it gives an indication of the magnitude of a parameter, but does not provide its exact value. Subjective knowledge is expressed by statements that contain vague terms, qualifications, probabilities, or judgmental data. Objects described by these vague statements are more difficult to fit into crisp sets. 

Now that the definition of a volatile liquid has been settled, the expert system could apply the rule. However, this approach is clearly unsatisfactory. The all-or-nothing crisp set that defines "volatile" does not allow for degrees of volatility. This conflicts with our common sense notion of volatility as a description, which changes smoothly from low-boiling liquids, like diethyl ether (boiling point = 34.6°C), which are widely accepted to be volatile, to materials like graphite or steel that are nonvolatile. If a human expert used the rule ... 

Within a fuzzy system, an inference engine works with fuzzy rules it takes input, part of which may be fuzzy, and generates output, some or all of which may be fuzzy. Although the role of a fuzzy system is to deal with uncertain data, the input is itself not necessarily fuzzy. For example, the data fed into the system might consist of the pH of a solution or the molecular weight of a compound, both of which can be specified with minimal uncertainty. In addition, the output that the system is required to produce is of more value if it is provided in a form that is crisp "Set the thermostat to 78°C" is more helpful to a scientist than "raise the temperature of the oven." Consequently, the fuzzy core of the inference engine is bracketed by one step that can turn crisp data into fuzzy data, and another that does the reverse. 

In each particular application of classical set theory as well as fuzzy set theory, all the sets of concern (classical or fuzzy) are subsets of a fixed set, which consists of all objects relevant to the applications. This set is called a uniuersal set and it is always denoted in this chapter by X. To distinguish classical (nonfuzzy) sets from fuzzy sets, the former are referred to as crisp sets. 

Fuzzy sets are defined on any given universal set by functions analogous to characteristic functions of crisp sets. These functions are called membership functions. To define a fuzzy set A on a given universal set X, the membership function of A assigns to each element jc of a number in the unit interval [0,1]. This number is viewed as the degree of membership of jc in A. 

Contrary to the symbolic role of numbers 1 and 0 in characteristic functions of crisp sets, numbers assigned to relevant objects by membership functions of fuzzy sets have a numerical significance. This significance is preserved when crisp sets are viewed (from the standpoint of fuzzy set theory) as special fuzzy sets. 

Among the most important concepts associated with fuzzy sets are the concepts of an a-cut and a strong a-cut. Given a fuzzy set A defined on X and a number a in the unit interval [0,1], the a-cut of A, denoted by °A, is the crisp set that consists of all elements of A whose membership degrees in A are greater than or equal to a that is,... 

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. 

In this interpretation, the possibility degree that the value of is in any given crisp set A is equal to the greatest possibility degree for all x A. That is, given a possibility distribution function rp, the associated possibility measure Pos is defined for all crisp subsets A of X via the formula... 

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. 

Can one understand fuzzy classical behavior in chemistry by specifying some strictly classical behavior at an appropriate limit (corresponding to crisp sets in fuzzy set theory) and giving quantum deviations therefrom for finitely many degrees of freedom or finite nuclear molecular masses (corresponding to proper fuzzy sets). 

There are, of course, possibilities of applying mathematical fuzzy set theory directly to the problems discussed here. The subsets 5 or used for the definition of the effective thermal states (62) and (63) should not be defined as crisp sets but as fuzzy sets. The respective compatibility functions of such fuzzy set definitions for and 5/j would assume very low values for the pure states T, which satisfy the condition... 

Nevertheless, an unsealed fuzzy Hausdorff metric provides important insight into the generalization of concepts originally proposed and proved for crisp sets to fuzzy sets, and some features of the proof can be utilized in the proof of the metric properties of the scaled fuzzy Hausdorff-type metric described here. For this purpose, first a proof is presented for the metric properties of an unsealed fuzzy Hausdorff metric, equivalent to the metric proposed by Puri and Ralescu, followed by a proof of the metric properties of an a-scaled fuzzy Hausdorff-type metric, suitable for fuzzy electron densities. 

Alternative symmetry deficiency measures of fuzzy sets are defined following the treatment of symmetry deficiency of ordinary subsets of finite n-dimensional Euclidean spaces, introduced earlier. To this end, we shall use certain concepts derived as generalizations of concepts in crisp set theory. 

A fuzzy set B is called an R-deficient set if B has none of the point symmetry elements of family R. However, by analogy with the case of crisp sets, it takes only infinitesimal distortions to lose a given symmetry element. Consequently, unless further restrictions are applied, the total mass difference between a fuzzy set of a specified symmetry and another fuzzy set that does not have this symmetry can be infinitesimal. As a result, i -deficient fuzzy sets and fuzzy R sets can be almost identical. Nevertheless, the actual symmetry deficiencies of fuzzy continua, such as formal molecular bodies represented by fuzzy clouds of electron densities, can be defined in terms of the deviations from their maximal R subsets and minimal R supersets, defined in subsequent text. 

Chirality, an important shape property of molecules, can be regarded as the lack of certain symmetry elements. Chirality measures are in fact measures of symmetry deficiency. These principles, originally used for crisp sets, also apply for fuzzy sets. Considering the case of three-dimensional chirality, the lacking point symmetry elements are reflection planes a and rotation-reflections 82 of even indices. Whereas the lacking symmetry elements can be of different nature in different dimensions, nevertheless, all the concepts, definitions, and procedures discussed in this section have straightforward generalizations for any finite dimension n. 

This FSNSM fs g is the real part of the complex fuzzy similarity-asymmetric shape tolerance measure fsast g. This measure is defined as fsast g = [fS( g, + fS(g, ]/2-P/[fS( g) - fS(g, ]/2 and is entirely analogous to the complex similarity-asymmetric shape tolerance measure sast g given for crisp sets in Eq. (135). The symmetric real part of fsast g is a measure of similarity, and the antisymmetric imaginary part of fsast g describes the asymmetry of shape tolerance. If the roles of fuzzy sets A and B are interchanged, then one obtains the complex conjugate (fsast g) of fsast g. 

XII. THE FUZZY AVERAGE OF CRISP SETS, THE FUZZY AVERAGE OF FUZZY SETS, THE CRISP AVERAGE OF CRISP SETS,... 

In all other cases, the fuzzy average Ff, of a family F of crisp sets is a genuine fuzzy set, since the average in definition (199) is over the necessarily binary membership function values for crisp sets, and if not all of the equations in Eq. (200) hold, for example, if... 

Consider a crisp or fuzzy subset A of the Euclidean space X, a (possibly approximate) symmetry element R, and the associated symmetry operator R. A fixed point of R is chosen as a reference point c e A, and a local Cartesian coordinate system of origin c is specified, with coordinate axes oriented according to the usual conventions with respect to the symmetry operator R, as described for crisp sets in Section XIII. 

The interpretation of these segments and the notation P for the convention used for the positioning of R with respect to set A and for the partitioning Ag, Aj,..., of the space A are the same as those used for crisp sets, described in Section XIII. 

It is obvious that any available spectral information can constrain the generation of an enormous number of structures, but its fuz2y character will greatly complicate the problem. Note that generators are created mostly for a simple enumeration of chemical structures. Therefore, they produce a crisp set of crisp structures. Each such structure is correlated to spectral information of a fuzzy nature. 

The one-to-one correspondence between the NMR signals and the carbon atoms and the assignment of possibility values through the membership function to each signal transforms the crisp set of vertices V from definition (3) of the graph into a fuzzy set forming a fuzzy molecular graph. 

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