Fuzzy set

In mathematics, fuzzy sets (aka uncertain sets) are somewhat like sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh[1] and Dieter Klaua[2] in 1965 as an extension of the classical notion of set. 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, which are used now in different areas, such as linguistics (De Cock, Bodenhofer & Kerre 2000), decision-making (Kuzmin 1982), and clustering (Bezdek 1978), 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.[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]

Definition

A fuzzy set is a pair ${\displaystyle (U,m)}$ where ${\displaystyle U}$ is a set and ${\displaystyle m\colon U\rightarrow [0,1]}$ a membership function. The reference set ${\displaystyle U}$ (sometimes denoted by ${\displaystyle \Omega }$ or ${\displaystyle X}$) is called universe of discourse, and for each ${\displaystyle x\in U,}$ the value ${\displaystyle m(x)}$ is called the grade of membership of ${\displaystyle x}$ in ${\displaystyle (U,m)}$. The function ${\displaystyle m=\mu (A)}$ is called the membership function of the fuzzy set ${\displaystyle A=(U,m)}$.

For a finite set ${\displaystyle U=\{x_{1},\dots ,x_{n}\},}$ the fuzzy set ${\displaystyle (U,m)}$ is often denoted by ${\displaystyle \{m(x_{1})/x_{1},\dots ,m(x_{n})/x_{n}\}.}$

Let ${\displaystyle x\in U.}$ Then ${\displaystyle x}$ is called

• not included in the fuzzy set ${\displaystyle (U,m)}$ if ${\displaystyle m(x)=0}$ (no member),
• fully included if ${\displaystyle m(x)=1}$ (full member),
• partially included if ${\displaystyle 0 (fuzzy member).[5]

The (crisp) set of all fuzzy sets on a universe ${\displaystyle U}$ is denoted with ${\displaystyle SF(U)}$ (or sometimes just ${\displaystyle F(U)}$).[6]

Crisp sets related to a fuzzy set

For any fuzzy set ${\displaystyle A=(U,m)}$ and ${\displaystyle \alpha \in [0,1]}$ the following crisp sets are defined:

• ${\displaystyle A^{\geq \alpha }=A_{\alpha }=\{x\in U\mid m(x)\geq \alpha \}}$ is called its α-cut (aka α-level set)
• ${\displaystyle A^{>\alpha }=A'_{\alpha }=\{x\in U\mid m(x)>\alpha \}}$ is called its strong α-cut (aka strong α-level set)
• ${\displaystyle S(A)=Supp(A)=A^{>0}=\{x\in U\mid m(x)>0\}}$ is called its support
• ${\displaystyle C(A)=Core(A)=A^{=1}=\{x\in U\mid m(x)=1\}}$ is called its core (or sometimes kernel ${\displaystyle Kern(A)}$).

Note that some authors understand 'kernel' in a different way, see below.

Other definitions

• A fuzzy set ${\displaystyle A=(U,m)}$ is empty (${\displaystyle A=\varnothing }$) iff (if and only if)
${\displaystyle \forall }$${\displaystyle x\in U:\mu _{A}(x)=m(x)=0}$
• Two fuzzy sets ${\displaystyle A}$ and ${\displaystyle B}$ are equal (${\displaystyle A=B}$) iff
${\displaystyle \forall x\in U:\mu _{A}(x)=\mu _{B}(x)}$
• A fuzzy set ${\displaystyle A}$ is included in a fuzzy set ${\displaystyle B}$ (${\displaystyle A\subseteq B}$) iff
${\displaystyle \forall x\in U:\mu _{A}(x)\leq \mu _{B}(x)}$
• For any fuzzy set ${\displaystyle A}$, any element ${\displaystyle x\in U}$ that satisfies
${\displaystyle \mu _{A}(x)=0.5}$
is called a crossover point.
• Given a fuzzy set A, any ${\displaystyle \alpha \in [0,1]}$, for which ${\displaystyle A^{=\alpha }=\{x\in U|\mu _{A}(x)=\alpha \}}$ is not empty, is called a level of A.
• The level set of A is the set of all levels α∈[0,1] representing distinct cuts. It is the target set (aka codomain) of ${\displaystyle \mu _{A}}$:
${\displaystyle \Lambda _{A}=\{\alpha \in [0,1]\mid A^{=\alpha }\neq \varnothing \}=\{\alpha \in [0,1]\mid {}}$${\displaystyle \exists }$${\displaystyle x\in {U}:\mu _{A}(x)=\alpha \}=\mu _{A}(U)}$
• For a fuzzy set ${\displaystyle A}$, its height is given by
${\displaystyle Hgt(A)=\sup\{\mu _{A}(x)\mid x\in {U}\}=\sup(\mu _{A}(U))}$
where ${\displaystyle \sup }$ denotes the supremum, which is known to exist because 1 is an upper bound. If U is finite, we can simply replace the supremum by the maximum.
• A fuzzy set ${\displaystyle A}$ is said to be normalized iff
${\displaystyle Hgt(A)=1}$
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 ${\displaystyle A}$ may be normalized with result ${\displaystyle {\tilde {A}}}$ by dividing the membership function of the fuzzy set by its height:
${\displaystyle \forall x\in {U}:\mu _{\tilde {A}}(x)=\mu _{A}(x)/Hgt(A)}$
Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum.
• For fuzzy sets ${\displaystyle A}$ of real numbers (U ⊆ ℝ) having a support with an upper and a lower bound, the width is defined as
${\displaystyle Width(A)=\sup(Supp(A))-\inf(Supp(A))}$
This does always exist for a bounded reference set U, including when U is finite.
In case that ${\displaystyle Supp(A)}$ is a finite or closed set, the width is just
${\displaystyle Width(A)=\max(Supp(A))-\min(Supp(A))}$
In the n-dimensional case (U ⊆ ℝn) the above can be replaced by the n-dimensional volume of ${\displaystyle Supp(A)}$.
In general, this can be defined given any measure on U, for instance by integration (e. g. Lebesgue integration) of ${\displaystyle Supp(A)}$.
• A real fuzzy set ${\displaystyle A}$ (U ⊆ ℝ) is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff
${\displaystyle \forall x,y\in {U},\forall \lambda \in [0,1]:\mu _{A}(\lambda {x}+(1-\lambda )y)\geq \min(\mu _{A}(x),\mu _{A}(y))}$.
Without loss of generality, we may take x≤y, which gives the equivalent formulation
${\displaystyle \forall z\in [x,y]:\mu _{A}(z)\geq \min(\mu _{A}(x),\mu _{A}(y))}$.
This definition can be extended to one for a general topological space U: we say the fuzzy set ${\displaystyle A}$ is convex when, for any subset Z of U, the condition
${\displaystyle \forall z\in Z:\mu _{A}(z)\geq \inf(\mu _{A}(\partial {Z}))}$
holds, where ${\displaystyle \partial {Z}}$ denotes the boundary of Z and ${\displaystyle f(X)=\{f(x)\mid x\in X\}}$ denotes the image of a set X (here ${\displaystyle \partial {Z}}$) under a function f (here ${\displaystyle \mu _{A}}$).

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 ${\displaystyle A}$, its complement ${\displaystyle \neg {A}}$ (sometimes denoted as ${\displaystyle A^{c}}$ or ${\displaystyle cA}$) is defined by the following membership function:
${\displaystyle \forall x\in {U}:\mu _{\neg {A}}(x)=1-\mu _{A}(x)}$.
• Let t be a t-norm, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets ${\displaystyle A,B}$, their intersection ${\displaystyle A\cap {B}}$ is defined by:
${\displaystyle \forall x\in {U}:\mu _{A\cap {B}}(x)=t(\mu _{A}(x),\mu _{B}(x))}$,
and their union ${\displaystyle A\cup {B}}$ is defined by:
${\displaystyle \forall x\in {U}:\mu _{A\cup {B}}(x)=s(\mu _{A}(x),\mu _{B}(x))}$.

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 by recursion.

• If the standard negator ${\displaystyle n(\alpha )=1-\alpha ,\alpha \in [0,1]}$ is replaced by another strong negator, the fuzzy set difference may be generalized by
${\displaystyle \forall x\in {U}:\mu _{\neg {A}}(x)=n(\mu _{A}(x))}$.
• 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 min 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 ${\displaystyle A}$ and ${\displaystyle \nu \in \mathbb {R} ^{+}}$ the ν-th power of A is defined by the membership function:
${\displaystyle \forall x\in {U}:\mu _{A^{\nu }}(x)=\mu _{A}(x)^{\nu }}$.

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

• For any fuzzy set ${\displaystyle A}$ the concentration ${\displaystyle CON(A)=A^{2}}$ is defined
${\displaystyle \forall x\in {U}:\mu _{CON(A)}(x)=\mu _{A^{2}}(x)=\mu _{A}(x)^{2}}$.

Of course, taking ${\displaystyle 0^{0}=1}$, we have ${\displaystyle A^{0}=U}$ and ${\displaystyle A^{1}=A}$.

• Given fuzzy sets ${\displaystyle A,B}$, the fuzzy set difference ${\displaystyle A\setminus B}$, also denoted ${\displaystyle A-B}$, may be defined straightforwardly via the membership function:
${\displaystyle \forall x\in {U}:\mu _{A\setminus {B}}(x)=t(\mu _{A}(x),n(\mu _{B}(x)))}$,
which means ${\displaystyle A\setminus B=A\cap \neg {B}}$, e. g.:
${\displaystyle \forall x\in {U}:\mu _{A\setminus {B}}(x)=\min(\mu _{A}(x),1-\mu _{B}(x))}$.[7][8]
Another proposal for a set difference could be:
${\displaystyle \forall x\in {U}:\mu _{A-{B}}(x)=\mu _{A}(x)-t(\mu _{A}(x),\mu _{B}(x))}$.[8]
• Proposals for symmetric fuzzy set differences have been done by Dubois and Prade (1980), either by taking the absolute value, giving
${\displaystyle \forall x\in {U}:\mu _{A\triangle {B}}(x)=|\mu _{A}(x)-\mu _{B}(x)|}$,
or by using a combination of just max, min, and standard negation, giving
${\displaystyle \forall x\in {U}:\mu _{A\triangle {B}}(x)=\max(\min(\mu _{A}(x),1-\mu _{B}(x)),\,min(\mu _{B}(x),1-\mu _{A}(x)))}$[8].
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).[8]
• 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 ${\displaystyle A,B}$ are disjoint iff

${\displaystyle \forall x\in {U}:\mu _{A}(x)=0\lor \mu _{B}(x)=0}$

which is equivalent to

${\displaystyle \nexists }$ ${\displaystyle x\in {U}:\mu _{A}(x)>0\land \mu _{B}(x)>0}$

and also equivalent to

${\displaystyle \forall x\in {U}:min(\mu _{A}(x),\mu _{B}(x))=0}$

We keep in mind that min/max is a t/s-norm pair, and any other will do the job here as well.

Fuzzy sets are disjoint, iff their supports are disjoint according to the standard definition for crisp sets.

For disjoint fuzzy sets ${\displaystyle A,B}$ any intersection will give ∅, and any union will give the same result, which is denoted as

${\displaystyle A{\dot {\cup }}B=A\cup B}$

with its membership function given by

${\displaystyle \forall x\in {U}:\mu _{A{\dot {\cup }}{B}}(x)=\mu _{A}(x)+\mu _{B}(x)}$

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

For disjoint fuzzy sets ${\displaystyle A,B}$ the following holds true:

${\displaystyle Supp(A{\dot {\cup }}B)=Supp(A)\,{\dot {\cup }}\,Supp(B)}$

This can be generalized to finite families of fuzzy sets as follows: Given a family ${\displaystyle A=(A_{i})_{i\in {I}}}$ of fuzzy sets with Index set I (e.g. I = {1,2,3,...n}). This family is (pairwise) disjoint iff

${\displaystyle \forall x\in {U}\,\exists i\in {I}:\mu _{A_{i}}(x)=0}$

A family of fuzzy sets ${\displaystyle A=(A_{i})_{i\in {I}}}$ is disjoint, iff the family of underlying supports ${\displaystyle Supp\circ A=(Supp(A_{i}))_{i\in {I}}}$ is disjoint in the standard sense for families of crisp sets.

Independend of the t/s-norm pair, intersection of a disjoint family of fuzzy sets will give ∅ again, while the union has no ambiguity:

${\displaystyle {\dot {\bigcup \limits _{i\in {I}}}}A_{i}=\bigcup _{i\in {I}}A_{i}}$

with its membership function given by

${\displaystyle \forall x\in {U}:\mu _{{\dot {\bigcup \limits _{i\in {I}}}}A_{i}}(x)=\sum _{i\in {I}}\mu _{A_{i}}(x)}$

Again only one of the summands is greater than zero.

For disjoint families of fuzzy sets ${\displaystyle A=(A_{i})_{i\in {I}}}$ the following holds true:

${\displaystyle Supp({\dot {\bigcup \limits _{i\in {I}}}}A_{i})={\dot {\bigcup \limits _{i\in {I}}}}Supp(A_{i})}$

Scalar Cardinality

For a fuzzy set ${\displaystyle A}$ with finite ${\displaystyle Supp(A)}$ (i. e. a 'finite fuzzy set'), its cardinality (aka scalar cardinality or sigma-count) is given by

${\displaystyle Card(A)=sc(A)=|A|=\sum _{x\in {U}}\mu _{A}(x)}$.

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

${\displaystyle RelCard(A)=||A||=sc(A)/|U|=|A|/|U|}$.

This can be generalized for the divisor to be an non-empty fuzzy set: For fuzzy sets ${\displaystyle A,G}$ with G ≠ ∅, we can define the relative cardinality by:

${\displaystyle RelCard(A,G)=sc(A|G)=sc(A\cap {G})/sc(G)}$,

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

• ${\displaystyle sc(G)>0}$ here.
• The result may depend on the specific intersection (t-norm) chosen.
• For ${\displaystyle G=U}$ the result is unambiguous and resembles the prior definition.

Distance and Similarity

For any fuzzy set ${\displaystyle A}$ the membership function ${\displaystyle \mu _{A}:U\to U}$ can be regarded as a family ${\displaystyle \mu _{A}=(\mu _{A}(x))_{x\in {U}}\in [0,1]^{U}}$. The latter is a metric space with several metrics ${\displaystyle d}$ known. A metric can be derived from a norm (vector norm) ${\displaystyle \|\,\|}$ via

${\displaystyle d(\alpha ,\beta )=\|\alpha -\beta \,\|}$.

For instance, if ${\displaystyle U}$ is finite, i. e. ${\displaystyle U=\{x_{1},x_{2},...x_{n}\}}$, such a metric may be defined by:

${\displaystyle d(\alpha ,\beta ):=max\{|\alpha (x_{i})-\beta (x_{i})|{\bigl \vert }i=1..n\}}$ where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are sequences of real numbes between 0 and 1.

For infinite ${\displaystyle U}$, 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:

${\displaystyle d(A,B):=d(\mu _{A},\mu _{B})}$,

which becomes in the above sample:

${\displaystyle d(A,B)=max\{|\mu _{A}(x_{i})-\mu _{B}(x_{i})|{\bigl \vert }i=1..n\}}$

Again for infinite ${\displaystyle U}$ 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 ${\displaystyle \varnothing }$ and ${\displaystyle U}$.

Similarity measures (here denoted by ${\displaystyle S}$) may then be derived from the distance, e. g. after a proposal by Koczy:

${\displaystyle S=1/(1+d(A,B))}$ if ${\displaystyle d(A,B)}$ is finite, ${\displaystyle 0}$ else,

or after Williams an Steele:

${\displaystyle S=exp(-\alpha {d(A,B)})}$ if ${\displaystyle d(A,B)}$ is finite, ${\displaystyle 0}$ else

where ${\displaystyle \alpha >0}$ is a steepness parameter and ${\displaystyle exp(x)=e^{x}}$.[6]

Another definition for interval valued (rather 'fuzzy') similarity measures ${\displaystyle \zeta }$ is provided by Beg and Ashraf as well.[6]

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 ${\displaystyle L}$ of a given kind; usually it is required that ${\displaystyle L}$ 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.[9] 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) ${\displaystyle A}$ is characterized by two functions:

1. ${\displaystyle \mu _{A}(x)}$ - degree of membership of x
2. ${\displaystyle \nu _{A}(x)}$ - degree of non-membership of x

with functions ${\displaystyle \mu _{A},\nu _{A}:U\mapsto [0,1]}$ with ${\displaystyle \forall x\in U:\mu _{A}(x)+\nu _{A}(x)\leq 1}$

This resembles a situation like some person denoted by ${\displaystyle x}$ voting

• for a proposal A (${\displaystyle \mu _{A}(x)=1,\nu _{A}(x)=0}$),
• against it (${\displaystyle \mu _{A}(x)=0,\nu _{A}(x)=1}$),
• or abstain from voting (${\displaystyle \mu _{A}(x)=\nu _{A}(x)=0}$).

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 provided. With ${\displaystyle D^{*}=\{(\alpha ,\beta )\in [0,1]^{2}\mid \alpha +\beta =1\}}$ and by combining both functions to ${\displaystyle (\mu _{A},\nu _{A}):U\to D^{*}}$ 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]: ${\displaystyle \mu _{A},\eta _{A},\nu _{A}}$, 'degree of positive membership', 'degree of neutral membership', and 'degree of negative membership' respectively and additional condition ${\displaystyle \forall x\in U:\mu _{A}(x)+\eta _{A}(x)+\nu _{A}(x)\leq 1}$ This expands the voting sample above by an additional possibility 'refusal of voting'.

With ${\displaystyle D^{*}=\{(\alpha ,\beta ,\gamma )\in [0,1]^{3}\mid \alpha +\beta +\gamma =1\}}$ and special 'picture fuzzy' negators, t- and s-norms this resembles just another type of L-fuzzy sets.[10][11]

Fuzzy logic

As an extension of the case of multi-valued logic, valuations (${\displaystyle \mu :{\mathit {V}}_{o}\to {\mathit {W}}}$) of propositional variables (${\displaystyle {\mathit {V}}_{o}}$) into a set of membership degrees (${\displaystyle {\mathit {W}}}$) 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.[12]

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."[13]

Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.

Fuzzy number and interval

A fuzzy number is a convex, normalized fuzzy set ${\displaystyle {\mathit {A}}\subseteq \mathbb {R} }$ of real numbers (U ⊆ ℝ) whose membership function is at least segmentally continuous[clarification needed] and has the functional value ${\displaystyle \mu _{A}(x)=1}$ at at least one element.[3] Because of the assumed convexity the maximum (of 1) is

• either an interval: fuzzy interval, its core is a crisp interval (mean interval) with lower bound
${\displaystyle \min \,C(A)=\min(\{x\in \mathbb {R} \mid \mu _{A}(x)=1\})}$
and upper bound
${\displaystyle \max \,C(A)=\max(\{x\in \mathbb {R} \mid \mu _{A}(x)=1\})}$.
• or unique: fuzzy number, its core is a singleton; the location of the maximum is
℩ C(A) = ℩${\displaystyle x\in \mathbb {R} :\mu _{A}(x)=1}$ (where ℩ reads as 'this');
which will assign a 'sharp' number to the fuzzy number, in addition to fuzzyness parameters like ${\displaystyle Width(A)}$.

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 ${\displaystyle {\mathit {A}}\subseteq \mathbb {R} }$ with a core interval, i. e. a mean interval whose elements possess the membership function value ${\displaystyle \mu _{A}(x)=1}$. 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.[14] Like crisp intervals, fuzzy intervals may reach infinity. The kernel ${\displaystyle K(A)=Kern(A)}$ of a fuzzy interval ${\displaystyle A}$ 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 ${\displaystyle \mathbb {R} }$ where ${\displaystyle \mu _{A}(x)}$ 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 components of category theory can be generalized to fuzzy sets. This approach which initiated in 1968 shortly after the introduction of fuzzy set theory[15] led to the development of "Goguen categories" in the 21st century.[16] [17] In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.[17][18]

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 fuzzyness for fuzzy sets of universe ${\displaystyle U}$ should fulfill the following conditions for all ${\displaystyle x\in U}$:

1. ${\displaystyle d(A)=0}$ if ${\displaystyle A}$ is a crisp set: ${\displaystyle \mu _{A}(x)\in \{0,\,1\}}$
2. ${\displaystyle d(A)}$ has a unique maximum iff ${\displaystyle \forall x\in U:\mu _{A}(x)=0.5}$
3. ${\displaystyle d(A)\geq d(B)}$ iff
${\displaystyle \mu _{a}(x)\leq \mu _{B}(x)}$ for ${\displaystyle \mu _{A}(x)\leq 0.5}$ and
${\displaystyle \mu _{a}(x)\geq \mu _{B}(x)}$ for ${\displaystyle \mu _{A}(x)\geq 0.5}$,
which means that B is 'crisper' than A.
1. ${\displaystyle d(\neg {A})=d(A)}$

In this case ${\displaystyle d(A)}$ is called the entropy of the fuzzy set A.

For finite ${\displaystyle U=\{x_{1},x_{2},...x_{n}\}}$ the entropy of a fuzzy set ${\displaystyle A}$ is given by

${\displaystyle d(A)=H(A)+H(\neg {A})}$,
${\displaystyle H(A)=-k\sum _{i=1}^{n}\mu _{A}(x_{i})\ln \mu _{A}(x_{i})}$

or just

${\displaystyle d(A)=-k\sum _{i=1}^{n}S(\mu _{A}(x_{i}))}$

where ${\displaystyle S(x)=H_{e}(x)}$ is Shannon's function (natural entropy function)

${\displaystyle S(\alpha )=-\alpha \ln \alpha -(1-\alpha )\ln(1-\alpha ),\alpha \in [0,1]}$

and ${\displaystyle k}$ is a constant depending on the measure unit and the logarithm base (here: e) used. Physical interpretation of k is the Boltzmann constant kB.

Let ${\displaystyle A}$ be a fuzzy set with a continuous membership function (fuzzy variable). Then

${\displaystyle H(A)=-k\int _{-\infty }^{\infty }\operatorname {Cr} \lbrace A\geq t\rbrace \ln \operatorname {Cr} \lbrace A\geq t\rbrace \,dt}$

and its entropy is

${\displaystyle d(A)=-k\int _{-\infty }^{\infty }S(\operatorname {Cr} \lbrace A\geq t\rbrace )\,dt}$

Extensions

There are many mathematical constructions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965, a lot of 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).

The diversity of such constructions and corresponding theories includes:

• interval sets (Moore, 1966),
• L-fuzzy sets (Goguen, 1967),
• flou sets (Gentilhomme, 1968),
• Boolean-valued fuzzy sets (Brown, 1971),
• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),
• set-valued sets (Chapin, 1974; 1975),
• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
• functions as generalizations of fuzzy sets and multisets (Lake, 1976),
• level fuzzy sets (Radecki, 1977)
• underdetermined sets (Narinyani, 1980),
• 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),
• named sets (Burgin, 1990),
• vague sets (Wen-Lung Gau and Buehrer, 1993),
• Q-sets (Gylys, 1994)
• α-level sets (Yao, 1997),
• genuine sets (Demirci, 1999),
• soft sets (Molodtsov, 1999),
• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)
• blurry sets (Smith, 2004)
• L-fuzzy rough sets (Radzikowska and Kerre, 2004),
• 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)
• bipolar fuzzy sets (Wen-Ran Zhang, 1998)
• multi-fuzzy sets (Sabu Sebastian, 2009)

While most of the above can be generally categorized as truth-based extensions to fuzzy sets, bipolar fuzzy set theory presents a philosophically and logically different, equilibrium-based generalization of fuzzy sets.[21][22][23]

Notes

1. ^ L. A. Zadeh (1965) "Fuzzy sets". Information and Control 8 (3) 338–353.
2. ^ Klaua, D. (1965) Über einen Ansatz zur mehrwertigen Mengenlehre. Monatsb. Deutsch. Akad. Wiss. Berlin 7, 859–876. A recent in-depth analysis of this paper has been provided by Gottwald, S. (2010). "An early approach toward graded identity and graded membership in set theory". Fuzzy Sets and Systems. 161 (18): 2369–2379. doi:10.1016/j.fss.2009.12.005.
3. ^ a b D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
4. ^ Lily R. Liang, Shiyong Lu, Xuena Wang, Yi Lu, Vinay Mandal, Dorrelyn Patacsil, and Deepak Kumar, "FM-test: A Fuzzy-Set-Theory-Based Approach to Differential Gene Expression Data Analysis", BMC Bioinformatics, 7 (Suppl 4): S7. 2006.
5. ^ AAAI Archived 2008-08-05 at the Wayback Machine
6. ^ a b c Ismat Beg, Samina Ashraf: Similarity measures for fuzzy sets, at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016
7. ^ Mamoni Dhar: Cardinality of Fuzzy Sets: An Overview, International Journal of Energy, Information and Communications Vol. 4, Issue 1, February 2013
8. ^ a b c d 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
9. ^ Goguen, Joseph A., 196, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
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