Information about Fuzzy Set
Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets have been introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set. In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — 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 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.
where 
is a set and 
. For each 
, 
is the grade of membership of 
. 
. If 
the fuzzy set can be denoted 
.
An element mapping to the value 0 means that the member is not included in the fuzzy set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.
Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure
; usually it is required that be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This generalization was first considered by Joseph Goguen (1967).
) of propositional variables (
) into a set of membership degrees (
) can be thought of as membership functions mapping predicates 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 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 fields of automated control and knowledge engineering, 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.
whose membership function is at least segmentally continuous and has the functional value 
at precisely one element.
This can be likened to the funfair game "guess your weight," where someone guesses the contestants weight, with closer guesses being more correct, and where the guesser "wins" if they guess near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).
with a mean interval whose elements possess the membership function value . As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous.
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
..... Click the link for more information.
Definition
A fuzzy set is a pair where is a set and . For each , is the grade of membership of . . If the fuzzy set can be denoted .
An element mapping to the value 0 means that the member is not included in the fuzzy set, 1 describes a fully included member. Values strictly between 0 and 1 characterize the fuzzy members.
Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure
; usually it is required that be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This generalization was first considered by Joseph Goguen (1967).
Fuzzy logic
As an extension of the case of multi-valued logic, valuations () of propositional variables () into a set of membership degrees () can be thought of as membership functions mapping predicates 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 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 fields of automated control and knowledge engineering, 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 number
A fuzzy number is a convex, normalized fuzzy set
whose membership function is at least segmentally continuous and has the functional value at precisely one element.
This can be likened to the funfair game "guess your weight," where someone guesses the contestants weight, with closer guesses being more correct, and where the guesser "wins" if they guess 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, normalized, at least segmentally continuous.
See also
External links
- Uncertainty model Fuzziness
- The Algorithm of Fuzzy Analysis
- Fuzzy Image Processing
- Zadeh's 1965 paper on Fuzzy Sets
References
- Goguen, Joseph A., 1967, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
- Gottwald, Siegfried, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0863802621
- Zadeh, Lotfi A.,
- 1965, "Fuzzy sets," Information and Control 8: 338–353.
- 1975, "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
- 1978, "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28.
Lotfali Askar Zadeh
Lotfali A. Zadeh in 2004
Born Februrary 12, 1921
Nationality Iranian
Field Mathematics
Institutions U.C.
..... Click the link for more information.
Lotfali A. Zadeh in 2004
Born Februrary 12, 1921
Nationality Iranian
Field Mathematics
Institutions U.C.
..... Click the link for more information.
SET may stand for:
..... Click the link for more information.
- Sanlih Entertainment Television, a television channel in Taiwan
- Secure electronic transaction, a protocol used for credit card processing,
..... Click the link for more information.
Set theory is the mathematical theory of sets, which represent collections of abstract objects. It encompasses the everyday notions, introduced in primary school, often as Venn diagrams, of collections of objects, and the elements of, and membership in, such collections.
..... Click the link for more information.
..... Click the link for more information.
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.
..... Click the link for more information.
..... Click the link for more information.
indicator function or a characteristic function is a function defined on a set that indicates membership of an element in a subset of .
The indicator function of a subset of a set is a function
defined as
..... Click the link for more information.
The indicator function of a subset of a set is a function
defined as
..... Click the link for more information.
Universal algebra (sometimes called General algebra) is the field of mathematics that studies the ideas common to all algebraic structures.
..... Click the link for more information.
Basic idea
From the point of view of universal algebra, an algebra (or algebraic structure) is a set A..... Click the link for more information.
In mathematical logic, a structure is an object that gives semantic meanings to the symbols in a logical language. The most common setting is with first-order languages, but structures for typed and higher-order languages are also important.
..... Click the link for more information.
..... Click the link for more information.
partially ordered set (or poset) formalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that describes, for certain pairs of elements in the set, the requirement that one
..... Click the link for more information.
..... Click the link for more information.
lattice is a partially ordered set (or poset) in which every pair of elements has a unique supremum (the elements' least upper bound; called their join) and an infimum (greatest lower bound; called their meet).
..... Click the link for more information.
..... Click the link for more information.
Joseph Amadee Goguen (28 June,1941 – 3 July,2006) was a computer science professor in the Department of Computer Science and Engineering at the University of California, San Diego, U.S., who helped develop the OBJ family of programming languages.
..... Click the link for more information.
..... Click the link for more information.
Multi-valued logics are logical calculi in which there are more than two truth values. Traditionally, logical calculi are two-valued—that is, there are only two possible truth values (i.e. truth and falsehood) for any proposition to take.
..... Click the link for more information.
..... Click the link for more information.
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.
..... Click the link for more information.
..... Click the link for more information.
First-order logic (FOL) is a formal deductive system used by mathematicians, philosophers, linguists, and computer scientists. It goes by many names, including: first-order predicate calculus (FOPC), the lower predicate calculus,
..... Click the link for more information.
..... Click the link for more information.
- For other uses, see Premise
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
..... Click the link for more information.
Engineering is the applied science of acquiring and applying knowledge to design, analysis, and/or construction of works for practical purposes. The American Engineers' Council for Professional Development, also known as ECPD,[1] (later ABET [2]
..... Click the link for more information.
..... Click the link for more information.
Automation (ancient Greek: = self dictated), roboticization [1] or industrial automation or numerical control is the use of control systems such as computers to control industrial machinery and processes, replacing human operators.
..... Click the link for more information.
..... Click the link for more information.
Knowledge engineering (KE) refers to the building, maintaining and development of knowledge-based systems. It has a great deal in common with software engineering, and is related to many computer science domains such as artificial intelligence, databases, data mining, expert
..... Click the link for more information.
..... Click the link for more information.
Fuzzy Logic may refer to:
Fuzzy logic
..... Click the link for more information.
- Fuzzy Logic (album), the debut album by the Super Furry Animals
- Fuzzy logic, an application of fuzzy set theory
- For the music album, see Fuzzy Logic (album)
Fuzzy logic
..... Click the link for more information.
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. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
..... Click the link for more information.
..... Click the link for more information.
The concept of a normalizing constant arises in probability theory and a variety of other areas of mathematics.
..... Click the link for more information.
Definition and examples
In probability theory, a normalizing constant..... Click the link for more information.
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.
..... Click the link for more information.
..... Click the link for more information.
funfair or simply fair (e.g. "county fair", "state fair") is a small to medium sized traveling fair primarily composed of amusement rides. Larger fairs or the permanent fairs of cities and seaside resorts may be called a fairground
..... Click the link for more information.
..... Click the link for more information.
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. For example, a solid cube is convex, but anything that is hollow or has a dent in it, for example, a crescent shape, is not convex.
..... Click the link for more information.
..... Click the link for more information.
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.
..... Click the link for more information.
..... Click the link for more information.
Fuzzy measure theory considers a number of special classes of measures, each of which is characterized by a special property. Some of the measures used in this theory are plausibility and belief measures, fuzzy set membership function and the classical probability measures.
..... Click the link for more information.
..... Click the link for more information.
Generically, an alternative set theory is an alternative mathematical approach to the concept of set. It is a proposed alternative to the standard set theory.
Some of the alternative set theories are:
..... Click the link for more information.
Some of the alternative set theories are:
- the theory of semisets;
- rough set theory;
..... Click the link for more information.
Defuzzification is the process of producing a quantifiable result in fuzzy logic. Typically, a fuzzy system will have a number of rules that transform a number of variables into a "fuzzy" result, that is, the result is described in terms of membership in fuzzy sets.
..... Click the link for more information.
..... Click the link for more information.
Fuzzy Logic may refer to:
Fuzzy logic
..... Click the link for more information.
- Fuzzy Logic (album), the debut album by the Super Furry Animals
- Fuzzy logic, an application of fuzzy set theory
- For the music album, see Fuzzy Logic (album)
Fuzzy logic
..... Click the link for more information.
A fuzzy set operation is an operation on fuzzy sets. These operations are generalization of crisp set operations. There is more than one possible generalization. The most widely used operations are called standard fuzzy set operations.
..... Click the link for more information.
..... Click the link for more information.
In the field of artificial intelligence, neuro-fuzzy refers to hybrids of artificial neural networks and fuzzy logic. Neuro-fuzzy hybridization results in a hybrid intelligent system that synergizes these two techniques by combining the human-like reasoning style of fuzzy systems
..... Click the link for more information.
..... Click the link for more information.
This article is copied from an article on Wikipedia.org - the free encyclopedia created and edited by online user community. The text was not checked or edited by anyone on our staff. Although the vast majority of the wikipedia encyclopedia articles provide accurate and timely information please do not assume the accuracy of any particular article. This article is distributed under the terms of GNU Free Documentation License.
Herod_Archelaus
