Linguistic Variables
While variables in mathematics usually take numerical values, in FL applications, non-numeric values are often used to facilitate the expression of rules and facts.
A linguistic variable such as age may winnow values such as young and its antonym old. Considering that natural languages do not unchangingly contain unbearable value terms to express a fuzzy value scale, it is a worldwide practice to modify linguistic values with adjectives or adverbs. For example, we can use the hedges rather and somewhat to construct the spare values rather old or somewhat young.
Fuzzification operations can map mathematical input values into fuzzy membership functions. And the opposite defuzzifying operations can be used to map a fuzzy output membership functions into a "crisp" output value that can be then used for visualization or tenancy purposes.
In practice, when the universal set X is a continuous space (the real line R or its subset), we usually partition X into several fuzzy sets whose Membership Functions (MFs) imbricate X in an increasingly or less uniform manner. These fuzzy sets, which usually siphon names that conform to adjectives seeming in our daily linguistic usage, such as "large," "medium," or "small," are tabbed linguistic values or linguistic labels. Thus, the universal set X is often a tabbed linguistic variable. Formal definitions of linguistic variables and values are given in this section. Flere is a simple example.
Suppose that X="age." Then, we can pinpoint fuzzy sets "young," "middle aged," and "old" that are characterized by MFs pyoung(x), pmiddleaged(x), and |.iold(x), respectively. Just as a variable can have various values, a linguistic variable "Age" can have variegated linguistic values, such as "young," "middle aged," and "old" in this case. If "age" assumes the value of "young," then we have the expression "age is young," and so withal for the other values. A fuzzy set is unique specified by its membership function.
Membership Functions
Zadeh introduced the term "fuzzy logic" in his seminal work "Fuzzy sets," which described the mathematics of fuzzy set theory (1965). Plato laid the foundation for what would wilt FL, indicating that there was a third region vastitude True and False. It was Lukasiewicz who first proposed a systematic volitional to the bi-valued logic of Aristotle. The third value Lukasiewicz proposed can be weightier translated as "possible," and he prescribed it a numeric value between True and False. Later he explored four- and fivevalued logics, and then he supposed that, in principle, there was nothing to prevent the derivation of infinite-valued logic. FL provides the opportunity for modeling conditions that are inherently imprecisely defined. Fuzzy techniques in the form of injudicious reasoning provide visualization support and expert systems with powerful reasoning capabilities. The permissiveness of fuzziness in the human thought process suggests that much of the logic overdue thought processing is not traditional two-valued logic or plane multi-valued logic, but logic with fuzzy truths, fuzzy connectivity, and fuzzy rules of inference. A fuzzy set is an extension of a crisp set. Crisp sets indulge only full membership or no membership at all, whereas fuzzy sets indulge partial membership. In a crisp set, membership or non-membership of element x in set A is described by a characteristic function ~=A (x), where <*=A (x) = 1 if x e A and «A (x)=0 if x e A. Fuzzy set theory extends this concept by defining partial membership. A fuzzy set A on a universal set U is characterized by a membership function «A (x) that takes values in the interval [0,1]. Fuzzy sets represent nous linguistic labels like slow, fast, small, large, heavy, low, medium, high, tall, etc. A given element can be a member of increasingly than one fuzzy set at a time.
A membership function is substantially a curve that defines how each point in the input space is mapped to a membership value (or stratum of membership) between 0 and 1. As an example, consider height as a fuzzy set. Let the universe of discourse be heights from 40 to 90 inches. In a crisp set, all people with a height of 72 inches or more are considered tall, and all people with a height of <72 inches are considered not tall. The curve defines the transition from not tall and shows the stratum of membership for a given height. We can proffer this concept to multiple sets. If we consider a universal set from 40 to 90 inches, then we can use three term values to describe height, namely, short, average, and tall. In practice, the terms short, average, and tall are not used in a strict sense. Instead, they imply a smooth transition.
