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.L. A. Zadeh (1965) "Fuzzy sets" . Information and Control 8 (3) 338–353.Klaua, D. (1965) . Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by 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 unit interval . 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.D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York. 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 where is a set (often required to be non-empty) 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 The (crisp) set of all fuzzy sets on a universe is denoted with (or sometimes just ).
Crisp sets related to a fuzzy set For any fuzzy set and 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 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 non-empty 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 non-empty 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 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 n-dimensional case the above can be replaced by the n-dimensional 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 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 t-norm, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets , their intersection is defined by: , and their union is defined by: .
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: and . If the standard negator 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:
For any fuzzy set and the ν-th 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.: N.R. Vemuri, A.S. Hareesh, M.S. Srinath: Set Difference and Symmetric Difference of Fuzzy Sets, in: Fuzzy Sets Theory and Applications 2014, Liptovský Ján, Slovak Republic
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 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 / 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 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/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 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 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 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 (t-norm) 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 2-norm) 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 .
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 of a given kind; usually it is required that 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 (IFS) is characterized by two functions: 1. – degree of membership of x 2. – degree of non-membership 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 s-norms can be defined. With and by combining both functions to 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]: , "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition