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 Dieter Klaua 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 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 (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 (often required to be non-empty) 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 ${\displaystyle 0<m(x)<1}$ (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)
• ${\displaystyle S(A)=\operatorname {Supp} (A)=A^{>0}=\{x\in U\mid m(x)>0\}}$ is called its support
• ${\displaystyle C(A)=\operatorname {Core} (A)=A^{=1}=\{x\in U\mid m(x)=1\}}$ is called its core (or sometimes kernel ${\displaystyle \operatorname {Kern} (A)}$).

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 ${\displaystyle A}$, any ${\displaystyle \alpha \in [0,1]}$, for which ${\displaystyle A^{=\alpha }=\{x\in U\mid \mu _{A}(x)=\alpha \}}$ is not empty, is called a level of A.
• The level set of A is the set of all levels ${\displaystyle \alpha \in [0,1]}$ representing distinct cuts. It is the image of ${\displaystyle \mu _{A}}$:

${\displaystyle \Lambda _{A}=\{\alpha \in [0,1]:A^{=\alpha }\neq \varnothing \}=\{\alpha \in [0,1]:{}}$${\displaystyle \exists }$${\displaystyle x\in U(\mu _{A}(x)=\alpha )\}=\mu _{A}(U)}$

• For a fuzzy set ${\displaystyle A}$, its height is given by

${\displaystyle \operatorname {Hgt} (A)=\sup\{\mu _{A}(x)\mid x\in U\}=\sup(\mu _{A}(U))}$

where ${\displaystyle \sup }$ denotes the supremum, which exists because ${\displaystyle \mu _{A}(U)}$ is non-empty and bounded above by 1. If U is finite, we can simply replace the supremum by the maximum.

• A fuzzy set ${\displaystyle A}$ is said to be normalized iff

${\displaystyle \operatorname {Hgt} (A)=1}$

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:

${\displaystyle \forall x\in U:\mu _{\tilde {A}}(x)=\mu _{A}(x)/\operatorname {Hgt} (A)}$

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) with bounded support, the width is defined as

${\displaystyle \operatorname {Width} (A)=\sup(\operatorname {Supp} (A))-\inf(\operatorname {Supp} (A))}$

In the case when ${\displaystyle \operatorname {Supp} (A)}$ is a finite set, or more generally a closed set, the width is just

${\displaystyle \operatorname {Width} (A)=\max(\operatorname {Supp} (A))-\min(\operatorname {Supp} (A))}$

In the n-dimensional case (U ⊆ ℝⁿ) the above can be replaced by the n-dimensional volume of ${\displaystyle \operatorname {Supp} (A)}$.

In general, this can be defined given any measure on U, for instance by integration (e.g. Lebesgue integration) of ${\displaystyle \operatorname {Supp} (A)}$.

• 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 recursively.

• 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 ${\displaystyle A}$ 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

${\displaystyle \forall x\in U:\mu _{CON(A)}(x)=\mu _{A^{2}}(x)=\mu _{A}(x)^{2}.}$

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]

Another proposal for a set difference could be:

${\displaystyle \forall x\in U:\mu _{A-{B}}(x)=\mu _{A}(x)-t(\mu _{A}(x),\mu _{B}(x)).}$[7]

• Proposals for symmetric fuzzy set differences have been made 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

${\displaystyle \forall x\in U:\mu _{A\triangle B}(x)=\max(\min(\mu _{A}(x),1-\mu _{B}(x)),\min(\mu _{B}(x),1-\mu _{A}(x))).}$[7]

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).[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 ${\displaystyle A,B}$ 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 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 ${\displaystyle A,B}$ 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:

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

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

${\displaystyle {\text{for all }}x\in U{\text{ there exists at most one }}i\in I{\text{ such that }}\mu _{A_{i}}(x)>0.}$

A family of fuzzy sets ${\displaystyle A=(A_{i})_{i\in I}}$ is disjoint, iff the family of underlying supports ${\displaystyle \operatorname {Supp} \circ A=(\operatorname {Supp} (A_{i}))_{i\in I}}$ 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:

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

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:

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

Scalar cardinality

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

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

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

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

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

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

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

• ${\displaystyle \operatorname {sc} (G)>0}$ 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 ${\displaystyle \mu _{A}:U\to [0,1]}$ 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:

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

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

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 and 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.[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 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 ${\displaystyle A}$: (${\displaystyle \mu _{A}(x)=1,\nu _{A}(x)=0}$),
• 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 defined. With ${\displaystyle D^{*}=\{(\alpha ,\beta )\in [0,1]^{2}:\alpha +\beta =1\}}$ 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 of "refusal of voting".

With ${\displaystyle D^{*}=\{(\alpha ,\beta ,\gamma )\in [0,1]^{3}:\alpha +\beta +\gamma =1\}}$ and special "picture fuzzy" negators, t- and s-norms this resembles just another type of L-fuzzy sets.[9][10]

Neutrosophic fuzzy sets

Some Key Developments in the Introduction of Fuzzy Set Concepts.[11]

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 ${\displaystyle \mu _{A}(x)}$ and another for non-membership ${\displaystyle \nu _{A}(x)}$. The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate ${\displaystyle i_{A}(x)}$. This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate ${\displaystyle i_{A}(x)}$ value can be particularly useful when one cannot be very confident on the membership or non-membership values for item x.[13] In summary, neutrosophic fuzzy sets are associated with the following functions:

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

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

3. ${\displaystyle i_{A}(x)}$ – 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 ${\displaystyle \mu _{A}(x)+\nu _{A}(x)\leq 1}$, 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 ${\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1}$, 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 ${\displaystyle \mu _{A}(x)+\nu _{A}(x)\leq 1}$ is not valid. However, the less restrictive condition of ${\displaystyle \mu _{A}(x)^{2}+\nu _{A}(x)^{2}\leq 1}$ may be suitable in more domains.[11][13]

Bipolar fuzzy sets

Bipolar fuzzy set theory is one of a few set theories recognized by Lotfi A. Zadeh, founder of fuzzy logic.[17] Bipolar fuzzy sets extend the membership of fuzzy sets from the unipolar lattice [0,1] to the bipolar Cartesian product lattice [-1,0]x[0,+1].[18][19] While fuzzy sets are truth-based, bipolar fuzzy sets are equilibrium-based. Bipolar fuzzy sets lead to bipolar logic and bipolar fuzzy logic which are dynamic systems.[20] While causality has been logically undefinable with truth-based logic in science and philosophy, bipolar fuzzy sets and logic provide logically definable causality that bridges the classical world to the quantum world.[21]

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.[22]

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."[23]

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 ${\displaystyle {\mathit {A}}\subseteq \mathbb {R} }$ of real numbers whose membership function is at least segmentally continuous 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 'the');

which will assign a 'sharp' number to the fuzzy number, in addition to fuzziness parameters like ${\displaystyle \operatorname {Width} (A)}$.

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.[24] Like crisp intervals, fuzzy intervals may reach infinity. The kernel ${\displaystyle K(A)=\operatorname {Kern} (A)}$ 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 component of category theory can be generalized to fuzzy sets. This approach, which began in 1968 shortly after the introduction of fuzzy set theory,[25] led to the development of Goguen categories in the 21st century.[26][27] In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.[27][28]

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 .

Entropy

A measure d of fuzziness 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 used (here we have used the natural base e). The physical interpretation of k is the Boltzmann constant k^B.

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

${\displaystyle d(A)=-k\int _{-\infty }^{\infty }S(\operatorname {Cr} \lbrace A\geq t\rbrace )\,dt.}$[29][30]

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 harvnb error: no target: CITEREFBurginChunihin1997 (help); Kerre 2001; Deschrijver and Kerre, 2003). Among all the extensions, bipolar fuzzy sets (Zhang 1994, 1998) as a unique ontological basis has led to equilibrium-based bipolar dynamic logic (BDL), bipolar dynamic fuzzy logic (BDFL), and bipolar quantum linear algebra (BQLA) with logically definable causality for quantum intelligence and mind-light-matter unity.[31][32][33]

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

1. L. A. Zadeh (1965) "Fuzzy sets" Archived 2015-08-13 at the Wayback Machine. 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. D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.
4. Liang, Lily R.; Lu, Shiyong; Wang, Xuena; Lu, Yi; Mandal, Vinay; Patacsil, Dorrelyn; Kumar, Deepak (2006). "FM-test: A fuzzy-set-theory-based approach to differential gene expression data analysis". BMC Bioinformatics. 7: S7. doi:10.1186/1471-2105-7-S4-S7. PMC 1780132. PMID 17217525.
5. "AAAI". Archived from the original on August 5, 2008.
6. 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. 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
8. Goguen, Joseph A., 196, "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18: 145–174
9. Bui Cong Cuong, Vladik Kreinovich, Roan Thi Ngan: A classification of representable t-norm operators for picture fuzzy sets, in: Departmental Technical Reports (CS). Paper 1047, 2016
10. Tridiv Jyoti Neog, Dusmanta Kumar Sut: Complement of an Extended Fuzzy Set, in: International Journal of Computer Applications (097 5–8887), Volume 29 No.3, September 2011
11. Yanase J, Triantaphyllou E (2019). "A Systematic Survey of Computer-Aided Diagnosis in Medicine: Past and Present Developments". Expert Systems with Applications. 138: 112821. doi:10.1016/j.eswa.2019.112821.
12. Smarandache, Florentin (1998). Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis. American Research Press. ISBN 978-1879585638.
13. Yanase J, Triantaphyllou E (2019). "The Seven Key Challenges for the Future of Computer-Aided Diagnosis in Medicine". International Journal of Medical Informatics. 129: 413–422. doi:10.1016/j.ijmedinf.2019.06.017. PMID 31445285.
14. Yager, Ronald R. (June 2013). "Pythagorean fuzzy subsets". 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). IEEE: 57–61. doi:10.1109/IFSA-NAFIPS.2013.6608375. ISBN 978-1-4799-0348-1. S2CID 36286152.
15. Yager, Ronald R (2013). "Pythagorean membership grades in multicriteria decision making". IEEE Transactions on Fuzzy Systems. 22 (4): 958–965. doi:10.1109/TFUZZ.2013.2278989. S2CID 37195356.
16. Yager, Ronald R. (December 2015). Properties and applications of Pythagorean fuzzy sets. Springer, Cham. pp. 119–136. ISBN 978-3-319-26302-1.
17. Zadeh, L.A. (2008). Fuzzy Logic. doi:10.4249/scholarpedia.1766.
18. Zhang, W.-R. (1994). "Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis". NAFIPS/IFIS/NASA '94. Proceedings of the First International Joint Conference of The North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige. IEEE. pp. 305–309. doi:10.1109/IJCF.1994.375115. ISBN 0-7803-2125-1.
19. Zhang, W.-R. (1998). "YinYang bipolar fuzzy sets". 1998 IEEE International Conference on Fuzzy Systems Proceedings. IEEE World Congress on Computational Intelligence (Cat. No.98CH36228). 1. IEEE. pp. 835–840. doi:10.1109/fuzzy.1998.687599. ISBN 078034863X.
20. Zhang, Wen-Ran; Zhang, Lulu (2004). "YinYang bipolar logic and bipolar fuzzy logic". Information Sciences. 165 (3–4): 265–287. doi:10.1016/j.ins.2003.05.010.
21. Zhang, Wen-Ran (2019). "The road from fuzzy sets to definable causality and bipolar quantum intelligence —To the memory of Lotfi A. Zadeh". Journal of Intelligent & Fuzzy Systems. 36 (4): 3019–3032. doi:10.3233/JIFS-172159.
22. Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., ISBN 978-0-86380-262-1
23. "The concept of a linguistic variable and its application to approximate reasoning," Information Sciences 8: 199–249, 301–357; 9: 43–80.
24. "Fuzzy sets as a basis for a theory of possibility," Fuzzy Sets and Systems 1: 3–28
25. J. A. Goguen "Categories of fuzzy sets: applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968
26. Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007 Springer ISBN 9781402061639
27. Michael Winter "Representation theory of Goguen categories" Fuzzy Sets and Systems Volume 138, Issue 1, 16 August 2003, Pages 85–126
28. Goguen, J.A., "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18(1):145–174, 1967
29. Xuecheng, Liu (1992). "Entropy, distance measure and similarity measure of fuzzy sets and their relations". Fuzzy Sets and Systems. 52 (3): 305–318. doi:10.1016/0165-0114(92)90239-Z.
30. Li, Xiang (2015). "Fuzzy cross-entropy". Journal of Uncertainty Analysis and Applications. 3. doi:10.1186/s40467-015-0029-5.
31. Zhang, W.-R.; Zhang, H. J.; Shi, Y.; Chen, S. S. (2009). "Bipolar Linear Algebra and YinYang-N-Element Cellular Networks for Equilibrium-Based Biosystem Simulation and Regulation". International Journal of Information Technology and Decision Making. 17 (4): 547–576. doi:10.1142/S0218339009002958.
32. Zhang, W.-R. (2018). "From Equilibrium-Based Business Intelligence to Information Conservational Quantum-Fuzzy Cryptography — A Cellular Transformation of Bipolar Fuzzy Sets to Quantum Intelligence Machinery". IEEE Trans. on Fuzzy Systems. 26 (2): 656–669. doi:10.1109/TFUZZ.2017.2687408.
33. Zhang, W.-R. (2021). "Ground-0 Axioms vs. First Principles and Second Law: From the Geometry of Light and Logic of Photon to Mind-Light-Matter Unity-AI&QI". IEEE/CAA J. Autom. Sinica. 8 (3): 534–553. doi:10.1109/JAS.2021.1003868.

• Alkhazaleh, S. and Salleh, A.R. Fuzzy Soft Multiset Theory, Abstract and Applied Analysis, 2012, article ID 350600, 20 p.
• Atanassov, K. T. (1983) Intuitionistic fuzzy sets, VII ITKR's Session, Sofia (deposited in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian)
• Atanasov, K. (1986) Intuitionistic Fuzzy Sets, Fuzzy Sets and Systems, v. 20, No. 1, pp. 87–96
• Baruah, Hemanta K. (2011) The Theory of Fuzzy Sets: Beliefs and Realities, International Journal of Energy, Information and Communications, Vol, 2, Issue 2, 1 – 22.
• Baruah, Hemanta K. (2012) An Introduction to the Theory of Imprecise Sets: the Mathematics of Partial Presence, International Journal of Computational and Mathematical Sciences, Vol. 2, No. 2, 110 – 124.
• Bezdek, J.C. (1978). "Fuzzy partitions and relations and axiomatic basis for clustering". Fuzzy Sets and Systems. 1 (2): 111–127. doi:10.1016/0165-0114(78)90012-X.
• Blizard, W.D. (1989) Real-valued Multisets and Fuzzy Sets, Fuzzy Sets and Systems, v. 33, pp. 77–97
• Brown, J.G. (1971) A Note on Fuzzy Sets, Information and Control, v. 18, pp. 32–39
• Brutoczki Kornelia: Fuzzy Logic (Diploma) – Although this script has a lot of odditiies and intracies due to its incompleteness, it may be used a template for exercise in removing these issues.
• Burgin, M. Theory of Named Sets as a Foundational Basis for Mathematics, in Structures in Mathematical Theories, San Sebastian, 1990, pp.  417–420
• Burgin M. and Chunihin, A. (1997) Named Sets in the Analysis of Uncertainty, in Methodological and Theoretical Problems of Mathematics and Information Sciences, Kiev, pp.  72–85
• Gianpiero Cattaneo and Davide Ciucci, "Heyting Wajsberg Algebras as an Abstract Environment Linking Fuzzy and Rough Sets" in J.J. Alpigini et al. (Eds.): RSCTC 2002, LNAI 2475, pp. 77–84, 2002. doi:10.1007/3-540-45813-1_10
• Chamorro-Martínez, J. et al.: A discussion on fuzzy cardinality and quantification. Some applications in image processing, SciVerse ScienceDirect: Fuzzy Sets and Systems 257 (2014) 85–101, 30 May 2013
• Chapin, E.W. (1974) Set-valued Set Theory, I, Notre Dame J. Formal Logic, v. 15, pp. 619–634
• Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp. 255–267
• Chris Cornelis, Martine De Cock and Etienne E. Kerre, [Intuitionistic fuzzy rough sets: at the crossroads of imperfect knowledge], Expert Systems, v. 20, issue 5, pp. 260–270, 2003
• Cornelis, C., Deschrijver, C., and Kerre, E. E. (2004) Implication in intuitionistic and interval-valued fuzzy set theory: construction, classification, application, International Journal of Approximate Reasoning, v. 35, pp. 55–95
• De Cock, Martine; Bodenhofer, Ulrich; Kerre, Etienne E. (1–4 October 2000). Modelling Linguistic Expressions Using Fuzzy Relations. Proceedings of the 6th International Conference on Soft Computing. Iizuka, Japan. pp. 353–360. CiteSeerX 10.1.1.32.8117.
• Demirci, M. (1999) Genuine Sets, Fuzzy Sets and Systems, v. 105, pp. 377–384
• Deschrijver, G.; Kerre, E.E. (2003). "On the relationship between some extensions of fuzzy set theory". Fuzzy Sets and Systems. 133 (2): 227–235. doi:10.1016/S0165-0114(02)00127-6.
• Didier Dubois, Henri M. Prade, ed. (2000). Fundamentals of fuzzy sets. The Handbooks of Fuzzy Sets Series. 7. Springer. ISBN 978-0-7923-7732-0.
• Feng F. Generalized Rough Fuzzy Sets Based on Soft Sets, Soft Computing, July 2010, Volume 14, Issue 9, pp 899–911
• Gentilhomme, Y. (1968) Les ensembles flous en linguistique, Cahiers Linguistique Theoretique Appliqee, 5, pp. 47–63
• Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp. 145–174
• Gottwald, S. (2006). "Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches". Studia Logica. 82 (2): 211–244. doi:10.1007/s11225-006-7197-8. S2CID 11931230.. Gottwald, S. (2006). "Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches". Studia Logica. 84: 23–50. doi:10.1007/s11225-006-9001-1. S2CID 10453751. preprint..
• Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, pp. 149–160.
• Grzymala-Busse, J. Learning from examples based on rough multisets, in Proceedings of the 2nd International Symposium on Methodologies for Intelligent Systems, Charlotte, NC, USA, 1987, pp. 325–332
• Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp. 9–31.
• Ulrich Höhle, Stephen Ernest Rodabaugh, ed. (1999). Mathematics of fuzzy sets: logic, topology, and measure theory. The Handbooks of Fuzzy Sets Series. 3. Springer. ISBN 978-0-7923-8388-8.
• Jahn, K. U. (1975) Intervall-wertige Mengen, Math.Nach. 68, pp. 115–132
• Kaufmann, Arnold. Introduction to the theory of fuzzy subsets. Vol. 2. Academic Pr, 1975.
• Kerre, E.E. (2001). "A first view on the alternatives of fuzzy set theory". In B. Reusch; K-H. Temme (eds.). Computational Intelligence in Theory and Practice. Heidelberg: Physica-Verlag. pp. 55–72. doi:10.1007/978-3-7908-1831-4_4. ISBN 978-3-7908-1357-9. Missing or empty |title= (help)
• George J. Klir; Bo Yuan (1995). Fuzzy sets and fuzzy logic: theory and applications. Prentice Hall. ISBN 978-0-13-101171-7.
• Kuzmin, V.B. (1982). "Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations" (in Russian). Nauka, Moscow.
• Lake, J. (1976) Sets, fuzzy sets, multisets and functions, J. London Math. Soc., II Ser., v. 12, pp. 323–326
• Meng, D., Zhang, X. and Qin, K. Soft rough fuzzy sets and soft fuzzy rough sets, 'Computers & Mathematics with Applications', v. 62, issue 12, 2011, pp. 4635–4645
• Miyamoto, S. Fuzzy Multisets and their Generalizations, in 'Multiset Processing', LNCS 2235, pp. 225–235, 2001
• Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No. 4/5, pp. 19–31
• Moore, R.E. Interval Analysis, New York, Prentice-Hall, 1966
• Nakamura, A. (1988) Fuzzy rough sets, 'Notes on Multiple-valued Logic in Japan', v. 9, pp. 1–8
• Narinyani, A.S. Underdetermined Sets – A new datatype for knowledge representation, Preprint 232, Project VOSTOK, issue 4, Novosibirsk, Computing Center, USSR Academy of Sciences, 1980
• Pedrycz, W. Shadowed sets: representing and processing fuzzy sets, IEEE Transactions on System, Man, and Cybernetics, Part B, 28, 103–109, 1998.
• Radecki, T. Level Fuzzy Sets, 'Journal of Cybernetics', Volume 7, Issue 3–4, 1977
• Radzikowska, A.M. and Etienne E. Kerre, E.E. On L-Fuzzy Rough Sets, Artificial Intelligence and Soft Computing – ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings; 01/2004
• Salii, V.N. (1965). "Binary L-relations". Izv. Vysh. Uchebn. Zaved. Matematika (in Russian). 44 (1): 133–145.
• Ramakrishnan, T.V., and Sabu Sebastian (2010) 'A study on multi-fuzzy sets', Int. J. Appl. Math. 23, 713–721.
• Sabu Sebastian and Ramakrishnan, T. V.(2010) Multi-fuzzy sets, Int. Math. Forum 50, 2471–2476.
• Sabu Sebastian and Ramakrishnan, T. V.(2011) Multi-fuzzy sets: an extension of fuzzy sets, Fuzzy Inf.Eng. 1, 35–43.
• Sabu Sebastian and Ramakrishnan, T. V.(2011) Multi-fuzzy extensions of functions, Advance in Adaptive Data Analysis 3, 339–350.
• Sabu Sebastian and Ramakrishnan, T. V.(2011) Multi-fuzzy extension of crisp functions using bridge functions, Ann. Fuzzy Math. Inform. 2 (1), 1–8
• Sambuc, R. Fonctions φ-floues: Application a l'aide au diagnostic en pathologie thyroidienne, Ph. D. Thesis Univ. Marseille, France, 1975.
• Seising, Rudolf: The Fuzzification of Systems. The Genesis of Fuzzy Set Theory and Its Initial Applications—Developments up to the 1970s (Studies in Fuzziness and Soft Computing, Vol. 216) Berlin, New York, [et al.]: Springer 2007.
• Smith, N.J.J. (2004) Vagueness and blurry sets, 'J. of Phil. Logic', 33, pp. 165–235
• Werro, Nicolas: Fuzzy Classification of Online Customers, University of Fribourg, Switzerland, 2008, Chapter 2
• Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp. 23–37
• Yao, Y.Y., Combination of rough and fuzzy sets based on α-level sets, in: Rough Sets and Data Mining: Analysis for Imprecise Data, Lin, T.Y. and Cercone, N. (Eds.), Kluwer Academic Publishers, Boston, pp. 301–321, 1997.
• Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1–4, 1998, pp. 227 – 242
• Zadeh, L. (1975) The concept of a linguistic variable and its application to approximate reasoning–I, Inform. Sci., v. 8, pp. 199–249
• Hans-Jürgen Zimmermann (2001). Fuzzy set theory—and its applications (4th ed.). Kluwer. ISBN 978-0-7923-7435-0.