[1][2] At the same time, Salii (1965) defined a more general kind of structure called an "L-relation", which he studied in an abstract algebraic context; fuzzy relations are special cases of L-relations when L is the unit interval [0, 1].
They are now used throughout fuzzy mathematics, having applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978).
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].
The fuzzy set theory can be used in a wide range of domains in which information is incomplete or imprecise, such as bioinformatics.
the following crisp sets are defined: Note that some authors understand "kernel" in a different way; see below.
Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.
By the definition of the t-norm, we see that the union and intersection are commutative, monotonic, associative, and have both a null and an identity element.
It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators: The case of exponent two is special enough to be given a name.
is disjoint in the standard sense for families of crisp sets.
Independent of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity: with its membership function given by Again only one of the summands is greater than zero.
with G ≠ ∅, we can define the relative cardinality by: which looks very similar to the expression for conditional probability.
Because fuzzy sets are unambiguously defined by their membership function, this metric can be used to measure distances between fuzzy sets on the same universe: which becomes in the above sample: Again for infinite
Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g.,
) may then be derived from the distance, e.g. after a proposal by Koczy: or after Williams and Steele: where
[citation needed] Sometimes, more general variants of the notion of fuzzy set are used, with membership functions taking values in a (fixed or variable) algebra or structure
These are usually called L-fuzzy sets, to distinguish them from those valued over the unit interval.
These kinds of generalizations were first considered in 1967 by Joseph Goguen, who was a student of Zadeh.
[8] A classical corollary may be indicating truth and membership values by {f, t} instead of {0, 1}.
For this situation, special "intuitive fuzzy" negators, t- and s-norms can be defined.
this situation resembles a special kind of L-fuzzy sets.
and special "picture fuzzy" negators, t- and s-norms this resembles just another type of L-fuzzy sets.
With these valuations, many-valued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.
[15] 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.
The core of this fuzzy number is a singleton; its location is: Fuzzy numbers can be likened to the funfair 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).
However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.
This approach, which began in 1968 shortly after the introduction of fuzzy set theory,[18] led to the development of Goguen categories in the 21st century.
[20][21] There are numerous mathematical extensions similar to or more general than fuzzy sets.
Since fuzzy sets were introduced in 1965 by Zadeh, many new mathematical constructions and theories treating imprecision, inaccuracy, vagueness, uncertainty and vulnerability have been developed.
Then and its entropy is There are many mathematical constructions similar to or more general than fuzzy sets.
Since fuzzy sets were introduced in 1965, many new mathematical constructions and theories treating imprecision, inexactness, ambiguity, and uncertainty have been developed.