In mathematics, fuzzy sets (a.k.a. uncertain 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.^{[1]}^{[2]} At the same time, Salii (1965) defined a more general kind of structure called an <i>L</i>relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics (De Cock, Bodenhofer & Kerre 2000), decisionmaking (Kuzmin 1982), and clustering (Bezdek 1978), are special cases of Lrelations when L is the unit interval [0, 1].
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 (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.^{[3]} 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.^{[4]}
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Definition
A fuzzy set is a pair where is a set (often required to be nonempty) and a membership function. The reference set (sometimes denoted by or ) is called universe of discourse, and for each the value is called the grade of membership of in . The function is called the membership function of the fuzzy set .
For a finite set the fuzzy set is often denoted by
Let . Then is called
 not included in the fuzzy set if (no member),
 fully included if (full member),
 partially included if (fuzzy member).^{[5]}
The (crisp) set of all fuzzy sets on a universe is denoted with (or sometimes just ).^{[6]}
For any fuzzy set and the following crisp sets are defined:
 is called its αcut (aka αlevel set)
 is called its strong αcut (aka strong αlevel 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
 A fuzzy set is empty () iff (if and only if)
 Two fuzzy sets and are equal () iff
 A fuzzy set is included in a fuzzy set () iff
 For any fuzzy set , any element that satisfies
 is called a crossover point.
 Given a fuzzy set , any , for which is not empty, is called a level of A.
 The level set of A is the set of all levels representing distinct cuts. It is the image of :
 For a fuzzy set , its height is given by
 where denotes the supremum, which exists because is nonempty and bounded above by 1. If U is finite, we can simply replace the supremum by the maximum.
 A fuzzy set is said to be normalized iff
 In the finite case, where the supremum is a maximum, this means that at least one element of the fuzzy set has full membership. A nonempty fuzzy set may be normalized with result 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.
 For fuzzy sets of real numbers (U ⊆ ℝ) with bounded support, the width is defined as
 In the case when is a finite set, or more generally a closed set, the width is just
 In the ndimensional case (U ⊆ ℝ^{n}) the above can be replaced by the ndimensional volume of .
 In general, this can be defined given any measure on U, for instance by integration (e.g. Lebesgue integration) of .
 A real fuzzy set (U ⊆ ℝ) is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff
 .
 Without loss of generality, we may take x ≤ y, which gives the equivalent formulation
 .
 This definition can be extended to one for a general topological space U: we say the fuzzy set is convex when, for any subset Z of U, the condition
 holds, where denotes the boundary of Z and denotes the image of a set X (here ) under a function f (here ).
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.
 For a given fuzzy set , its complement (sometimes denoted as or ) is defined by the following membership function:
 .
 Let t be a tnorm, and s the corresponding snorm (aka tconorm). Given a pair of fuzzy sets , their intersection is defined by:
 ,
 and their union is defined by:
 .
By the definition of the tnorm, 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.
 If the standard negator is replaced by another strong negator, the fuzzy set difference 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 tnorms.
 The fuzzy intersection is not idempotent in general, because the standard tnorm min is the only one which has this property. Indeed, if the arithmetic multiplication is used as the tnorm, 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 mth power of a fuzzy set, which can be canonically generalized for noninteger exponents in the following way:
 For any fuzzy set and the νth power of is defined by the membership function:
The case of exponent two is special enough to be given a name.
 For any fuzzy set the concentration is defined
Taking , we have and
 Given fuzzy sets , the fuzzy set difference , also denoted , may be defined straightforwardly via the membership function:
 which means , e. g.:
 ^{[7]}
 Another proposal for a set difference could be:
 ^{[7]}
 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 max, min, and standard negation, giving
 ^{[7]}
 Axioms for definition of generalized symmetric differences analogous to those for tnorms, tconorms, and negators have been proposed by Vemur et al. (2014) with predecessors by Alsina et al. (2005) and Bedregal et al. (2009).^{[7]}
 In contrast to crisp sets, averaging operations can also be defined for fuzzy sets.
Disjoint fuzzy sets
In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets: Two fuzzy sets are disjoint iff
which is equivalent to
and also equivalent to
We keep in mind that min/max is a t/snorm 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.
For disjoint fuzzy 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.
For disjoint fuzzy sets the following holds true:
This can be generalized to finite families of fuzzy sets as follows: Given a family of fuzzy sets with index set I (e.g. I = {1,2,3,...,n}). This family is (pairwise) disjoint iff
A family of fuzzy sets is disjoint, iff the family of underlying supports is disjoint in the standard sense for families of crisp sets.
Independent of the t/snorm 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.
For disjoint families of fuzzy sets the following holds true:
Scalar cardinality
For a fuzzy set with finite support (i.e. a "finite fuzzy set"), its cardinality (aka scalar cardinality or sigmacount) is given by
 .
In the case that U itself is a finite set, the relative cardinality is given by
 .
This can be generalized for the divisor to be a nonempty fuzzy set: For fuzzy sets with G ≠ ∅, we can define the relative cardinality by:
 ,
which looks very similar to the expression for conditional probability. Note:
 here.
 The result may depend on the specific intersection (tnorm) chosen.
 For the result is unambiguous and resembles the prior definition.
Distance and similarity
For any fuzzy set the membership function can be regarded as a family . The latter is a metric space with several metrics known. A metric can be derived from a norm (vector norm) via
 .
For instance, if is finite, i.e. , such a metric may be defined by:
 where and are sequences of real numbers between 0 and 1.
For infinite , 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:
 ,
which becomes in the above sample:
 .
Again for infinite the maximum must be replaced by a supremum. Other distances (like the canonical 2norm) may diverge, if infinite fuzzy sets are too different, e.g., and .
Similarity measures (here denoted by ) may then be derived from the distance, e.g. after a proposal by Koczy:
 if is finite, else,
or after Williams and Steele:
 if is finite, else
where is a steepness parameter and .^{[6]}
Another definition for interval valued (rather 'fuzzy') similarity measures is provided by Beg and Ashraf as well.^{[6]}
Lfuzzy 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 of a given kind; usually it is required that be at least a poset or lattice. These are usually called Lfuzzy 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.^{[8]} 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 (IFS) is characterized by two functions:
 1. – degree of membership of x
 2. – degree of nonmembership of x
with functions with .
This resembles a situation like some person denoted by voting
 for a proposal : (),
 against it: (),
 or abstain from voting: ().
After all, we have a percentage of approvals, a percentage of denials, and a percentage of abstentions.
For this situation, special "intuitive fuzzy" negators, t and snorms can be defined. With and by combining both functions to this situation resembles a special kind of Lfuzzy 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, 1]: , "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition This expands the voting sample above by an additional possibility of "refusal of voting".
With and special "picture fuzzy" negators, t and snorms this resembles just another type of Lfuzzy sets.^{[9]}^{[10]}
Neutrosophic fuzzy sets
The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets.^{[11]}
Neutrosophic fuzzy sets were introduced by Smarandache in 1998.^{[12]} Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership and another for nonmembership . The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate . This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate value can be particularly useful when one cannot be very confident on the membership or nonmembership values for item x.^{[13]} In summary, neutrosophic fuzzy sets are associated with the following functions:
 1. —degree of membership of x
 2. —degree of nonmembership of x
 3. —degree of indeterminate value of x
Pythagorean fuzzy sets
The other extension of IFS is what is known as Pythagorean fuzzy sets. Pythagorean fuzzy sets are more flexible than IFSs. IFSs are based on the constraint , which can be considered as too restrictive in some occasions. This is why Yager proposed the concept of Pythagorean fuzzy sets. Such sets satisfy the constraint , which is reminiscent of the Pythagorean theorem.^{[14]}^{[15]}^{[16]} Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of is not valid. However, the less restrictive condition of may be suitable in more domains.^{[11]}^{[13]}
Fuzzy logic
As an extension of the case of multivalued 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, manyvalued logic can be extended to allow for fuzzy premises from which graded conclusions may be drawn.^{[17]}
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."^{[18]}
Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.
Fuzzy number and only number
A fuzzy number^{[19]} is a fuzzy set that satisfies all the following conditions:
 A is normalised;
 A is a convex set;
 ;
 The membership function is at least segmentally continuous.
If these conditions are not satisfied, then A is not a fuzzy number. The core of this fuzzy number is a singleton; its location is:
When the condition about the uniqueness of is not fulfilled, then the fuzzy set is characterised as a fuzzy interval.^{[19]} The core of this fuzzy interval is a crisp interval with:
 .
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).
The kernel of a fuzzy interval is defined as the 'inner' part, without the 'outbound' parts where the membership value is constant ad infinitum. In other words, the smallest subset of where 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,^{[20]} led to the development of Goguen categories in the 21st century.^{[21]}^{[22]} In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in Lfuzzy sets.^{[22]}^{[23]}
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^{[citation needed]}.
Entropy
A measure d of fuzziness for fuzzy sets of universe should fulfill the following conditions for all :
 if is a crisp set:
 has a unique maximum iff
 which means that B is "crisper" than A.
In this case is called the entropy of the fuzzy set A.
For finite the entropy of a fuzzy set is given by
 ,
or just
where is Shannon's function (natural entropy function)
and 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 k^{B}.
Let be a fuzzy set with a continuous membership function (fuzzy variable). Then
and its entropy is
 ^{[24]}^{[25]}
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 (Burgin & Chunihin 1997 ; Kerre 2001; Deschrijver and Kerre, 2003).
See also
References
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