Fuzzy set

In mathematics, fuzzy sets (a.k.a. uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh and 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, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of "L"-relations 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 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.
Definition.
A fuzzy set is a pair formula_1 where formula_2 is a set (often required to be non-empty) and formula_3 a membership function. 
The reference set formula_2 (sometimes denoted by formula_5 or formula_6) is called universe of discourse, and for each formula_7 the value formula_8 is called the grade of membership of formula_9 in formula_10. 
The function formula_11 is called the membership function of the fuzzy set formula_12.
For a finite set formula_13 the fuzzy set formula_1 is often denoted by formula_15
Let formula_16. Then formula_9 is called 
The (crisp) set of all fuzzy sets on a universe formula_2 is denoted with formula_20 (or sometimes just formula_21).
Crisp sets related to a fuzzy set.
For any fuzzy set formula_22 and formula_23 the following crisp sets are defined:
Note that some authors understand "kernel" in a different way; see below.
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.
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.
The case of exponent two is special enough to be given a name.
Taking formula_95, we have formula_96 and formula_97
Disjoint fuzzy sets.
In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:
Two fuzzy sets formula_81 are disjoint iff
which is equivalent to
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.
For disjoint fuzzy sets formula_81 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 formula_81 the following holds true:
This can be generalized to finite families of fuzzy sets as follows:
Given a family formula_117 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 formula_117 is disjoint, iff the family of underlying supports formula_120 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.
For disjoint families of fuzzy sets formula_117 the following holds true:
Scalar cardinality.
For a fuzzy set formula_33 with finite support formula_63 (i.e. a "finite fuzzy set"), its cardinality (aka scalar cardinality or sigma-count) 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 non-empty fuzzy set: For fuzzy sets formula_129 with "G" ≠ ∅, we can define the relative cardinality by:
which looks very similar to the expression for conditional probability.
Note:
Distance and similarity.
For any fuzzy set formula_33 the membership function formula_134 can be regarded as a family formula_135. The latter is a metric space with several metrics formula_136 known. A metric can be derived from a norm (vector norm) formula_137 via
For instance, if formula_2 is finite, i.e. formula_140, such a metric may be defined by: 
For infinite formula_2, 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 formula_2 the maximum must be replaced by a supremum. Other distances (like the canonical 2-norm) may diverge, if infinite fuzzy sets are too different, e.g., formula_148 and formula_2.
Similarity measures (here denoted by formula_150) may then be derived from the distance, e.g. after a proposal by Koczy:
or after Williams and Steele:
where formula_157 is a steepness parameter and formula_158.
Another definition for interval valued (rather 'fuzzy') similarity measures formula_159 is provided by Beg and Ashraf as well.
"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 formula_160 of a given kind; usually it is required that formula_160 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 and Baruah. An intuitionistic fuzzy set (IFS) formula_33 is characterized by two functions:
with functions formula_165 with formula_166
This resembles a situation like some person denoted by formula_9 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 s-norms can be defined. With formula_172 and by combining both functions to formula_173 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, 1]: formula_174, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition formula_175
This expands the voting sample above by an additional possibility of "refusal of voting".
With formula_176 and special "picture fuzzy" negators, t- and s-norms this resembles just another type of "L"-fuzzy sets.
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.
Neutrosophic fuzzy sets were introduced by Smarandache in 1998. Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership formula_163 and another for non-membership formula_164. The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate formula_179. This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate formula_179 value can be particularly useful when one cannot be very confident on the membership or non-membership values for item "x". In summary, neutrosophic fuzzy sets are associated with the following functions:
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 formula_184, 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 formula_185, which is reminiscent of the Pythagorean theorem. Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of formula_184 is not valid. However, the less restrictive condition of formula_185 may be suitable in more domains.
Fuzzy logic.
As an extension of the case of multi-valued logic, valuations (formula_188) of propositional variables (formula_189) into a set of membership degrees (formula_190) 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 and only number.
A fuzzy number is a convex, normalized fuzzy set formula_191 of real numbers whose membership function is at least segmentally continuous and has the functional value formula_192 at at least one element. Because of the assumed convexity the maximum (of 1) 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).
A fuzzy interval is a fuzzy set formula_191 with a core interval, i.e. a mean interval whose elements possess the membership function value formula_192.
The latter means that fuzzy intervals are normalized fuzzy sets. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally continuous. Like crisp intervals, fuzzy intervals may reach infinity. The kernel formula_199 of a fuzzy interval formula_33 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 formula_201 where formula_163 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.
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" with "R" .
Entropy.
A measure "d" of fuzziness for fuzzy sets of universe formula_2 should fulfill the following conditions for all formula_16:
In this case formula_208 is called the entropy of the fuzzy set "A".
For finite formula_140 the entropy of a fuzzy set formula_33 is given by
or just
where formula_220 is Shannon's function (natural entropy function)
and formula_222 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 formula_33 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 (; ; Deschrijver and Kerre, 2003).