Membership characteristic function

Fuzzy logic is based on the generalization of theory of sets characteristic function that Zadeh defined as membership function , //U), (Zadeh, 1965)... 

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. 

A formal classification theory for such terms, which are neither very exact nor very inexact, was introduced by Zadeh, who, among others, saw the need for generalizing the theory of sets to admit the possibility of unsharp boundaries between classes of imprecisely defined objects. Whereas the characteristic function can only have a value of 0 or 1 (in harmony with the Aristotelian logic of the excluded middle), membership in a fuzzy set is a matter of degree. This degree (or grade) of membership (ix) may have any value in the continuum of the real interval [0,1] in the special case where the value of /x is restricted to 0 or 1, the degree of membership becomes the characteristic function. 

In classical set theory, the containment of an elementto a subset A of the universe of discourse X is described by a characteristic function. It is called membership function, m x). A membership value of 1 is assigned an element x, that is, contained in a set A. If X is not an element of the set A, a membership value of zero results ... 

A cortunon or crisp subset A of a given universe X,A QX, of elements is defined by specifying for every element x X of the universe whether it is a member of A(x e A) or it is not a member of A(x A). In terms of classical logic the notation X A has to be translated into x belongs to A is true otherwise the statement is false. A classical set can be described in different ways, i.e., by defining the member elements by a characteristic function, in which 1 indicates membership and 0 nonmembership ... 

In common, fuzzy sets are described by the generalized characteristic function, the membership function, p.(x). Thus, a fuzzy set A that is defined over a universe of discourse X is characterized by the generalized characteristic function pa x) ... 

Classical crisp sets contain objects that satisfy precise properties of membership, contrarily the fuzzy sets contain objects that satisfy imprecise properties of membership, i.e. membership of an object in a fuzzy set is not a matter of affirmation or denial, but rather a matter of a degree of belongingness. Suppose we have an exhaustive collection of individual elements x, which makes up a universe of discourse X. Further, various combinations of the individual elements make up a crisp set, say A. For set A, an element x in the universe X is either a member of A or not. Mathematically, the membership of element x in set zl can be expressed by the characteristic function ... 

Zadeh has observed that P( ) can be viewed as the expected value of the characteristic function that defines the set E (Zadeh (1987)). By analogy, he defines the probability of the fuzzy set A as the expected value of the membership function for A ... 

The principle of applying fuzzy logic to matching of spectra is that, given a sample spectrum and a collection of reference spectra, in a first step the reference spectra are unified and fuzzed, i.e., around each characteristic line at a certain wavenumber k, a certain fuzzy interval [/ o - Ak, + Afe] is laid. The resulting fuzzy set is then intersected with the crisp sample spectrum. A membership function analogous to the one in Figure 9-25 is applied. If a line of the sample spec-... 

The body surface chemicals of insects contain a large proportion of CHCs, including straight-chain, methyl-, dimethyl-, trimethyl-branched and unsaturated components. They function as important components in chemical communication (Howard, 1993). Ant species, especially, rely much on CHCs as chemical communication to inform the membership of colonies, caste, sex and other physiological status. The CHC components are different among species (see related chapters), and the mixture profiles of CHCs are characteristically different among colonies, but rather resemble the individual worker ants in each colony. Since colony members usually share similar CHC profiles in any ant species,... 

Fuzzy logic control calculations are executed by using both membership functions of the inputs and outputs and a set of rules called a rule base, as shown in Fig. 16.21. Typical membership functions for the inputs, e and deldt, are shown in Fig. 16.23, where it is assumed that these inputs have identical membership functions with the following characteristics three linguistic variables which are negative (N), positive P), and zero (Z) with trapezoidal, triangular and trapezoidal membership function forms respectively. Input variables e and deldt have been scaled so that the membership functions overlap for the range from -1 to +1. Furthermore, Fig. 16.24 shows the membership functions of the output Aw(r), which are... 

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