Universe of discourse

The central concept of fuzzy set theory is that the membership function /i, like probability theory, can have a value of between 0 and 1. In Figure 10.3, the membership function /i has a linear relationship with the x-axis, called the universe of discourse U. This produces a triangular shaped fuzzy set. 

Let the fuzzy set medium temperature be called fuzzy set M. If an element u of the universe of discourse U lies within fuzzy set M, it will have a value of between 0 and 1. This is expressed mathematically as... 

When the universe of discourse is discrete and finite, fuzzy set M may be expressed as... 

An important aspect of fuzzy logic is the ability to relate sets with different universes of discourse. Consider the relationship... 

Figure 10.9 assumes that the output window contains seven fuzzy sets with the same linguistic labels as the input fuzzy sets. If the universe of discourse for the control signal u(t) is 9, then the output window is as shown in Figure 10.10. 

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 . 

The x-axis in a plot of a membership function represents the universe of discourse. This is the complete range of values that the independent variable can take the y-axis is the membership value of the fuzzy set. 

Sets are very general mathematical objects that are used in many branches of mathematics. Here the focus is on finite sets, that is, sets with a finite set of elements. A key concept in set theory is that of the universal set, U, sometimes called the universe of discourse, which is an unordered collection of n elements x1,x2, , xk,.. . , xn and is given by... 

Remark 3. Although the number of elements known to exist in the universe is finite, it is seldom necessary to consider more than a few of them in a particular universe of discourse. Thus large tracts of organic chemistry require only the three elements, carbon, hydrogen and oxygen. 

Consider the module consisting of all lists of the form (ej, e2 2,..., r 5Bt) where the e, are integers and, ..., is a set of chemical elements sufficient for the universe of discourse. This may consist of atomic species, indecomposable molecules or radicals. We note in passing that this is the direct sum of the t submodules for which... 

We conclude that KS orbitals seem to be just as suitable, if not better, for qualitative MO theoretical considerations than other orbitals, e.g., HF orbitals. The KS orbitals offer the advantage, in particular over semiempirical orbitals, but also over HF, that they are connected in an interesting way with the exact wavefunction and with exact energetics. So the MO-theoretical analysis put forward in the next section deals with energetic contributions that sum up to the exact or, with the present state of the art in density functionals, at least accurate interaction energy. The KS model offers an MO-theoretical universe of discourse in which molecular energetics can be interpreted in terms of considerations that until now were necessarily inaccurate and qualitative. Is this MO-... 

This section is based on discussions and correspondence, during over two decades, with the seven of the other living investigators who have created molecular periodic systems (and of course on the author s own experience). All of them share the belief that the universe of discourse is comprehensible and that our modes of thinking and communicating about it are adequate. 

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 ... 

Fuzzy sets are a generalization of conventional set theory. They were introduced by Zadeh [106] as a mathematical way to represent vagueness in everyday fife. A formal definition of fuzzy sets that has been presented by many researchers is following a fuzzy set A is a subset of the universe of discourse X that admits partial membership. The fuzzy set A is defined as the ordered pair A = (x, where... 

As mentioned earlier, the common feature of these models is that they look at MIC as a whole without particularly focusing on a given type of microorganism/macroorganism. In this respect, they tend to be more generalized in their universe of discourse in the sense that MIC is taken as a possibility among other corrosion processes and factors that can happen in a system. 

The fuzzy set theory is an outgrowth of the classical set theory. First, recall the classical set theory, which views the world as either black or white. Let X be the universe of discourse and x be its elements. According to the classical set theory, crisp set of X is defined by the characteristic function (x) of set A. 

The range of possible values of a linguistic variable represents the variable s universe of discourse. For example, the universe of discourse of the linguistic variable completion date might have the range between 1 and 10 days, and include fuzzy subsets such as early, normal and late. 

Figure 2.9 shows an application of hedges (very). The universe of discourse -men s heights - consists of five fuzzy sets very short, short, average, tallandvery tall. For example, a man 180 cm tall is a member of the tall set with a degree of membership of 0.5 and a member of the very tall set with a degree of membership of 0.2. 

Fuzzy intersection Fuzzy intersection is the fuzzy operation for creating the intersection of fuzzy sets A and B on the universe of discourse X, which can be obtained as ... 

FIGURE 3.1 The Universe of Discourse X is partitioned into 10 indiscernibility classes by the set of attributes A,. See Section 3.2.1 of the text for further discussion, three of which lie within the set 5 being approximated. 

Universal Set Universe of discourse or fixed set from which subsets are formed written U. 

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