Fuzzy set

# Fuzzy set

Overview
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh
Lotfali Askar Zadeh , better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artifical intelligence researcher and professor emeritus of computer science at the University of California, Berkeley...

and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, the membership of elements in a set is assessed in binary terms according to a bivalent condition
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...

— an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function
Membership function (mathematics)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy...

valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory
Discussion

Encyclopedia
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced simultaneously by Lotfi A. Zadeh
Lotfali Askar Zadeh , better known as Lotfi A. Zadeh, is a mathematician, electrical engineer, computer scientist, artifical intelligence researcher and professor emeritus of computer science at the University of California, Berkeley...

and Dieter Klaua in 1965 as an extension of the classical notion of set. In classical set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...

, the membership of elements in a set is assessed in binary terms according to a bivalent condition
Principle of bivalence
In logic, the semantic principle of bivalence states that every declarative sentence expressing a proposition has exactly one truth value, either true or false...

— an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function
Membership function (mathematics)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy...

valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1. In fuzzy set theory, classical bivalent sets are usually called crisp sets. The fuzzy set theory
can be used in a wide range of domains in which information is incomplete or imprecise,
such as bioinformatics
Bioinformatics
Bioinformatics is the application of computer science and information technology to the field of biology and medicine. Bioinformatics deals with algorithms, databases and information systems, web technologies, artificial intelligence and soft computing, information and computation theory, software...

.

Fuzzy sets can be applied, for example, to the field of genealogical research
Genealogy
Genealogy is the study of families and the tracing of their lineages and history. Genealogists use oral traditions, historical records, genetic analysis, and other records to obtain information about a family and to demonstrate kinship and pedigrees of its members...

. When an individual is searching in vital records such as birth records
Birth certificate
A birth certificate is a vital record that documents the birth of a child. The term "birth certificate" can refer to either the original document certifying the circumstances of the birth or to a certified copy of or representation of the ensuing registration of that birth...

for possible ancestors, the researcher must contend with a number of issues that could be encapsulated in a membership function. Looking for an ancestor named John Henry Pittman, who you think was born in (probably eastern) Tennessee circa 1853 (based on statements of his age in later censuses
United States Census
The United States Census is a decennial census mandated by the United States Constitution. The population is enumerated every 10 years and the results are used to allocate Congressional seats , electoral votes, and government program funding. The United States Census Bureau The United States Census...

, and a marriage record in Knoxville), what is the likelihood that a particular birth record for "John Pittman" is your John Pittman? What about a record in a different part of Tennessee for "J.H. Pittman" in 1851? (It has been suggested by Thayer Watkins that Zadeh's ethnicity is an example of a fuzzy set)

## Definition

A fuzzy set is a pair where is a set and .

For each , is called the grade of membership of in . For a finite set , the fuzzy set is often denoted by .

Let . Then is called not included in the fuzzy set if , is called fully included if , and is called a fuzzy member if .
The set is called the support of and the set is called its kernel.

Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra
Algebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...

or structure
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....

of a given kind; usually it is required that be at least a poset or lattice
Lattice (order)
In mathematics, a lattice is a partially ordered set in which any two elements have a unique supremum and an infimum . Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities...

. These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. These kinds of generalizations were first considered in 1967 by Joseph Goguen
Joseph Goguen
Joseph Amadee Goguen was a computer science professor in the Department of Computer Science and Engineering at the University of California, San Diego, USA, who helped develop the OBJ family of programming languages. He was author of A Categorical Manifesto and founder and Editor-in-Chief of the...

, who was a student of Zadeh.

## Fuzzy logic

As an extension of the case of multi-valued logic
Multi-valued logic
In logic, a many-valued logic is a propositional calculus in which there are more than two truth values. Traditionally, in Aristotle's logical calculus, there were only two possible values for any proposition...

, valuations () of propositional variable
Propositional variable
In mathematical logic, a propositional variable is a variable which can either be true or false...

s () into a set of membership degrees () can be thought of as membership functions
Membership function (mathematics)
The membership function of a fuzzy set is a generalization of the indicator function in classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy...

mapping predicates
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

into fuzzy sets (or more formally, into an ordered set of fuzzy pairs, called a fuzzy relation). With these valuations, many-valued logic can be extended to allow for fuzzy premises
Premises
Premises are land and buildings together considered as a property. This usage arose from property owners finding the word in their title deeds, where it originally correctly meant "the aforementioned; what this document is about", from Latin prae-missus = "placed before".In this sense, the word is...

from which graded conclusions may be drawn.

This extension is sometimes called "fuzzy logic in the narrow sense" as opposed to "fuzzy logic in the wider sense," which originated in the engineering
Engineering
Engineering is the discipline, art, skill and profession of acquiring and applying scientific, mathematical, economic, social, and practical knowledge, in order to design and build structures, machines, devices, systems, materials and processes that safely realize improvements to the lives of...

fields of automated
Automation
Automation is the use of control systems and information technologies to reduce the need for human work in the production of goods and services. In the scope of industrialization, automation is a step beyond mechanization...

control and knowledge engineering
Knowledge engineering
Knowledge engineering was defined in 1983 by Edward Feigenbaum, and Pamela McCorduck as follows:At present, it refers to the building, maintaining and development of knowledge-based systems...

, and which encompasses many topics involving fuzzy sets and "approximated reasoning."

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...

.

## Fuzzy number

A fuzzy number is a convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

, normalized
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...

fuzzy set
whose membership function is at least segmentally continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

and has the functional value at precisely one element.

This can be likened to the funfair
Funfair
A funfair or simply "fair" is a small to medium sized travelling show primarily composed of stalls and other amusements. Larger fairs such as the permanent fairs of cities and seaside resorts might be called a fairground, although technically this should refer to the land where a fair is...

game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

## Fuzzy interval

A fuzzy interval is an uncertain set with a mean interval whose elements possess the membership function value . As in fuzzy numbers, the membership function must be convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...

, normalized
Normalizing constant
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.-Definition and examples:In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g.,...

, at least segmentally continuous
Continuous function
In mathematics, a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be "discontinuous". A continuous function with a continuous inverse function is called "bicontinuous".Continuity of...

.

## Fuzzy relation equation

The fuzzy relation equation is an equation of the form A · R = B, where A and B are fuzzy sets, R is a fuzzy relation, and A · R stands for the composition of A with R.