In mathematics, fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, 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. They are now used throughout fuzzy mathematics, having applications in areas such as linguistics , decision-making , and clustering .

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. Fuzzy sets generalize classical sets, since the indicator functions (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes 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.

Definition

is a set (often required to be non-empty) is called universe of discourse is called the grade is called the membership function.

is called

• not included if (no member),

• fully included if (full member),

• partially included if

).

Crisp sets related to a fuzzy set

the following crisp sets are defined:

• is called its α-cut (aka α-level set)

• is called its strong α-cut (aka strong α-level set)

• is called its support

• is called its core (or sometimes kernel).

Note that some authors understand "kernel" in a different way; see below.

Other definitions

• is empty) iff (if and only if)

  1. \forall

  are equal) iff

  is included) iff

  that satisfies

• is called a crossover point.

• is not empty, is called a level of A.

• The level set representing distinct cuts. It is the image:

  1. \exist

  , its height is given by

• denotes the supremum is non-empty and bounded above by 1. If U is finite, we can simply replace the supremum by the maximum.

• is said to be normalized iff

• by dividing the membership function of the fuzzy set by its height:

• Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum.

• with bounded support, the width is defined as

• is a finite set, or more generally a closed set, the width is just

• In the n the above can be replaced by the n.

• In general, this can be defined given any measure on U, for instance by integration (e.g. Lebesgue integration.

• is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff

  1. .

  Without loss of generality, we may take x ≤ y, which gives the equivalent formulation

  1. .

  This definition can be extended to one for a general topological space U is convex when, for any subset Z of U, the condition

• denotes the boundary of Z denotes the image of a set X) under a function f).

Fuzzy set operations

Although the complement of a fuzzy set has a single most common definition, the other main operations, union and intersection, do have some ambiguity.

• , its complement) is defined by the following membership function:

  1. .

  Let t be a t-norm, their intersection is defined by:

  1. ,

  and their union is defined by:

  1. .

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. For the intersection, these are ∅ and U, respectively, while for the union, these are reversed. However, the union of a fuzzy set and its complement may not result in the full universe U, and the intersection of them may not give the empty set ∅. Since the intersection and union are associative, it is natural to define the intersection and union of a finite family of fuzzy sets recursively. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators:

• .

• is replaced by another strong negator, the fuzzy set difference (defined below) may be generalized by

• The triple of fuzzy intersection, union and complement form a De Morgan Triplet. That is, De Morgan's laws extend to this triple.

• Examples for fuzzy intersection/union pairs with standard negator can be derived from samples provided in the article about t-norms.

  The fuzzy intersection is not idempotent in general, because the standard t-norm is the only one which has this property. Indeed, if the arithmetic multiplication is used as the t-norm, the resulting fuzzy intersection operation is not idempotent. That is, iteratively taking the intersection of a fuzzy set with itself is not trivial. It instead defines the m-th power of a fuzzy set, which can be canonically generalized for non-integer exponents in the following way:

• the is defined by the membership function:

The case of exponent two is special enough to be given a name.

• the concentration is defined

  , the fuzzy set difference, may be defined straightforwardly via the membership function:

• , e. g.:

• Another proposal for a set difference could be:

• Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the absolute value, giving

• or by using a combination of just , , and standard negation, giving

• Axioms for definition of generalized symmetric differences analogous to those for t-norms, t-conorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).

• In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.

Disjoint fuzzy sets

are disjoint iff

which is equivalent to

1. \nexists

and also equivalent to

We keep in mind that / is a t/s-norm pair, and any other will work here as well.

Fuzzy sets are disjoint if and only if their supports are disjoint according to the standard definition for crisp sets.

any intersection will give ∅, and any union will give the same result, which is denoted as

with its membership function given by

Note that only one of both summands is greater than zero.

the following holds true:

of fuzzy sets with index set I (e.g. I = {1,2,3,...,n}). This family is (pairwise) disjoint iff

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.

the following holds true:

Scalar cardinality

(i.e. a "finite fuzzy set"), its cardinality (aka scalar cardinality or sigma-count) is given by

1. .

In the case that U itself is a finite set, the relative cardinality is given by

1. .

with G ≠ ∅, we can define the relative cardinality by:

1. ,

which looks very similar to the expression for conditional probability. Note:

• here.

• The result may depend on the specific intersection (t-norm) chosen.

• the result is unambiguous and resembles the prior definition.

Distance and similarity

. The latter is a metric space known. A metric can be derived from a norm via

1. .

, such a metric may be defined by:

1. are sequences of real numbers between 0 and 1.

, the maximum can be replaced by a supremum. 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:

1. ,

which becomes in the above sample:

1. .

.

) may then be derived from the distance, e.g. after a proposal by Koczy:

1. else,

or after Williams and Steele:

1. else

.

L-fuzzy sets

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 be at least a poset or lattice. 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, who was a student of Zadeh. A classical corollary may be indicating truth and membership values by {f,t} instead of {0,1}.

An extension of fuzzy sets has been provided by Atanassov. An intuitionistic fuzzy set is characterized by two functions:

1. – degree of membership of x

2. – degree of non-membership of x

.

voting

• ),

• ),

• ).

After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.

this situation resembles a special kind of L-fuzzy sets.

Once more, this has been expanded by defining picture fuzzy sets (PFS) as follows: A PFS A is characterized by three functions mapping U to [0, This expands the voting sample above by an additional possibility of "refusal of voting".

and special "picture fuzzy" negators, t- and s-norms this resembles just another type of L-fuzzy sets.

Pythagorean fuzzy sets

, which is reminiscent of the Pythagorean theorem. may be suitable in more domains.

Fuzzy logic

As an extension of the case of multi-valued logic) of propositional variables) 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 that satisfies all the following conditions:

• is normalised;

• is a convex set;

• achieves the value 1 at least once;

• is at least segmentally continuous.

is not a fuzzy number. 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).

is constant outside of it, is defined as the kernel.

However, there are other concepts of fuzzy numbers and intervals as some authors do not insist on convexity.

Fuzzy categories

The use of set membership as a key component of category theory can be generalized to fuzzy sets. This approach, which began in 1968 shortly after the introduction of fuzzy set theory, led to the development of Goguen categories in the 21st century. In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.

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. Some of these constructions and theories are extensions of fuzzy set theory, while others attempt to mathematically model inaccuracy/vagueness and uncertainty in a different way. The diversity of such constructions and corresponding theories includes:

• Fuzzy Sets (Zadeh, 1965)

• interval sets (Moore, 1966),

• L-fuzzy sets (Goguen, 1967),

• flou sets (Gentilhomme, 1968),

• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),

• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),

• level fuzzy sets (Radecki, 1977)

• rough sets (Pawlak, 1982),

• intuitionistic fuzzy sets (Atanassov, 1983),

• fuzzy multisets (Yager, 1986),

• intuitionistic L-fuzzy sets (Atanassov, 1986),

• rough multisets (Grzymala-Busse, 1987),

• fuzzy rough sets (Nakamura, 1988),

• real-valued fuzzy sets (Blizard, 1989),

• vague sets (Wen-Lung Gau and Buehrer, 1993),

• α-level sets (Yao, 1997),

• shadowed sets (Pedrycz, 1998),

• neutrosophic sets (NSs) (Smarandache, 1998),

• bipolar fuzzy sets (Wen-Ran Zhang, 1998),

• genuine sets (Demirci, 1999),

• soft sets (Molodtsov, 1999),

• complex fuzzy set (2002),

• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)

• L-fuzzy rough sets (Radzikowska and Kerre, 2004),

• multi-fuzzy sets (Sabu Sebastian, 2009),

• generalized rough fuzzy sets (Feng, 2010)

• rough intuitionistic fuzzy sets (Thomas and Nair, 2011),

• soft rough fuzzy sets (Meng, Zhang and Qin, 2011)

• soft fuzzy rough sets (Meng, Zhang and Qin, 2011)

• soft multisets (Alkhazaleh, Salleh and Hassan, 2011)

• fuzzy soft multisets (Alkhazaleh and Salleh, 2012)

• pythagorean fuzzy set (Yager, 2013),

• picture fuzzy set (Cuong, 2013),

• spherical fuzzy set (Mahmood, 2018).

Fuzzy relation equation

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

Entropy

A measure d:

1. , which means that A is "crisper" than B.

is called the entropy of the fuzzy set A.

For finite is given by

1. ,

or just

is Shannon's function (natural entropy function)

is a constant depending on the measure unit and the logarithm base used (here we have used the natural base e). The physical interpretation of k is the Boltzmann constant kB.

be a fuzzy set with a continuous membership function (fuzzy variable). Then

and its entropy is

Extensions

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. Some of these constructions and theories are extensions of fuzzy set theory, while others try to mathematically model imprecision and uncertainty in a different way.

See also

• Alternative set theory

• Defuzzification

• Fuzzy concept

• Fuzzy mathematics

• Fuzzy set operations

• Fuzzy subalgebra

• Interval finite element

• Linear partial information

• Multiset

• Neuro-fuzzy

• Rough fuzzy hybridization

• Rough set

• Sørensen similarity index

• Type-2 fuzzy sets and systems

• Uncertainty

References

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