Fuzzy set

A fuzzy set is a pair $(U,\; m)$ where $U$ is a set (often required to be non-empty) and $m\backslash colon\; U\; \backslash rightarrow\; [0,1]$ a membership function.

The reference set $U$ (sometimes denoted by $\backslash Omega$ or $X$) is called universe of discourse, and for each $x\backslash in\; U,$ the value $m(x)$ is called the grade of membership of $x$ in $(U,m)$.

The function $m\; =\; \backslash mu\_A$ is called the membership function of the fuzzy set $A\; =\; (U,\; m)$.

For a finite set $U=\backslash \{x\_1,\backslash dots,x\_n\backslash \},$ the fuzzy set $(U,\; m)$ is often denoted by $\backslash \{m(x\_1)/x\_1,\backslash dots,m(x\_n)/x\_n\backslash \}.$

Let $x\; \backslash in\; U$. Then $x$ is called

• not included in the fuzzy set $(U,m)$ if {{nowrap|$m(x)\; =\; 0$}} (no member),
• fully included if {{nowrap|$m(x)\; =\; 1$}} (full member),
• partially included if {{nowrap| (fuzzy member).{{Cite web|url=http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic|archive-url=https://web.archive.org/web/20080805071058/http://www.aaai.org/aitopics/pmwiki/pmwiki.php/AITopics/FuzzyLogic|url-status=dead|title=AAAI|archive-date=August 5, 2008}}}}

The (crisp) set of all fuzzy sets on a universe $U$ is denoted with $SF(U)$ (or sometimes just $F(U)$).{{cn|date=December 2024}}

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

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

Note that some authors understand "kernel" in a different way; see below.

• A fuzzy set $A\; =\; (U,m)$ is empty ($A\; =\; \backslash varnothing$) iff (if and only if)

::$\backslash forall$$x\; \backslash in\; U:\; \backslash mu\_A(x)\; =\; m(x)\; =\; 0$

• Two fuzzy sets $A$ and $B$ are equal ($A\; =\; B$) iff

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; =\; \backslash mu\_B(x)$

• A fuzzy set $A$ is included in a fuzzy set $B$ ($A\; \backslash subseteq\; B$) iff

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; \backslash le\; \backslash mu\_B(x)$

• For any fuzzy set $A$, any element $x\; \backslash in\; U$ that satisfies

::$\backslash mu\_A(x)\; =\; 0.5$

:is called a crossover point.

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

::$\backslash Lambda\_A\; =\; \backslash \{\backslash alpha\; \backslash in\; [0,1]\; :\; A^\{=\backslash alpha\}\; \backslash ne\; \backslash varnothing\backslash \}\; =\; \backslash \{\backslash alpha\; \backslash in\; [0,\; 1]\; :\; \{\}$$\backslash exist$$x\; \backslash in\; U(\backslash mu\_A(x)\; =\; \backslash alpha)\backslash \}\; =\; \backslash mu\_A(U)$

• For a fuzzy set $A$, its height is given by

::$\backslash operatorname\{Hgt\}(A)\; =\; \backslash sup\; \backslash \{\backslash mu\_A(x)\; \backslash mid\; x\; \backslash in\; U\backslash \}\; =\; \backslash sup(\backslash mu\_A(U))$

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

• A fuzzy set $A$ is said to be normalized iff

::$\backslash operatorname\{Hgt\}(A)\; =\; 1$

: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 $A$ may be normalized with result $\backslash tilde\{A\}$ by dividing the membership function of the fuzzy set by its height:

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

:Besides similarities this differs from the usual normalization in that the normalizing constant is not a sum.

• For fuzzy sets $A$ of real numbers $(U\; \backslash subseteq\; \backslash mathbb\{R\})$ with bounded support, the width is defined as

::$\backslash operatorname\{Width\}(A)\; =\; \backslash sup(\backslash operatorname\{Supp\}(A))\; -\; \backslash inf(\backslash operatorname\{Supp\}(A))$

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

::$\backslash operatorname\{Width\}(A)\; =\; \backslash max(\backslash operatorname\{Supp\}(A))\; -\; \backslash min(\backslash operatorname\{Supp\}(A))$

:In the n-dimensional case $(U\; \backslash subseteq\; \backslash mathbb\{R\}^n)$ the above can be replaced by the n-dimensional volume of $\backslash operatorname\{Supp\}(A)$.

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

• A real fuzzy set $A\; (U\; \backslash subseteq\; \backslash mathbb\{R\})$ is said to be convex (in the fuzzy sense, not to be confused with a crisp convex set), iff

::$\backslash forall\; x,y\; \backslash in\; U,\; \backslash forall\backslash lambda\backslash in[0,1]:\; \backslash mu\_A(\backslash lambda\{x\}\; +\; (1-\backslash lambda)y)\; \backslash ge\; \backslash min(\backslash mu\_A(x),\backslash mu\_A(y))$.

: Without loss of generality, we may take x ≤ y, which gives the equivalent formulation

::$\backslash forall\; z\; \backslash in\; [x,y]:\; \backslash mu\_A(z)\; \backslash ge\; \backslash min(\backslash mu\_A(x),\backslash mu\_A(y))$.

: This definition can be extended to one for a general topological space U: we say the fuzzy set $A$ is convex when, for any subset Z of U, the condition

::$\backslash forall\; z\; \backslash in\; Z:\; \backslash mu\_A(z)\; \backslash ge\; \backslash inf(\backslash mu\_A(\backslash partial\; Z))$

: holds, where $\backslash partial\; Z$ denotes the boundary of Z and $f(X)\; =\; \backslash \{f(x)\; \backslash mid\; x\; \backslash in\; X\backslash \}$ denotes the image of a set X (here $\backslash partial\; Z$) under a function f (here $\backslash mu\_A$).

{{main|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 $A$, its complement $\backslash neg\{A\}$ (sometimes denoted as $A^c$ or $cA$) is defined by the following membership function:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{\backslash neg\{A\}\}(x)\; =\; 1\; -\; \backslash mu\_A(x)$.

• Let t be a t-norm, and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets $A,\; B$, their intersection $A\backslash cap\{B\}$ is defined by:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\backslash cap\{B\}\}(x)\; =\; t(\backslash mu\_A(x),\backslash mu\_B(x))$,

:and their union $A\backslash cup\{B\}$ is defined by:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\backslash cup\{B\}\}(x)\; =\; s(\backslash mu\_A(x),\backslash mu\_B(x))$.

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. It is noteworthy that the generally accepted standard operators for the union and intersection of fuzzy sets are the max and min operators:

• $\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\backslash cup\{B\}\}(x)\; =\; \backslash max(\backslash mu\_A(x),\backslash mu\_B(x))$ and $\backslash mu\_\{A\backslash cap\{B\}\}(x)\; =\; \backslash min(\backslash mu\_A(x),\backslash mu\_B(x))$.{{cite journal|last1=Bellman|first1=Richard|last2=Giertz|first2=Magnus|date=1973|title=On the analytic formalism of the theory of fuzzy sets|journal=Information Sciences|volume=5|pages=149–156|doi=10.1016/0020-0255(73)90009-1}}

• If the standard negator $n(\backslash alpha)\; =\; 1\; -\; \backslash alpha,\; \backslash alpha\; \backslash in\; [0,\; 1]$ is replaced by another strong negator, the fuzzy set difference (defined below) may be generalized by

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{\backslash neg\{A\}\}(x)\; =\; n(\backslash mu\_A(x)).$

• 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 {{math|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 $A$ and $\backslash nu\; \backslash in\; \backslash R^+$ the ν-th power of $A$ is defined by the membership function:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A^\{\backslash nu\}\}(x)\; =\; \backslash mu\_\{A\}(x)^\{\backslash nu\}.$

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

• For any fuzzy set $A$ the concentration $CON(A)\; =\; A^2$ is defined

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

Taking $0^0\; =\; 1$, we have $A^0\; =\; U$ and $A^1\; =\; A.$

• Given fuzzy sets $A,\; B$, the fuzzy set difference $A\; \backslash setminus\; B$, also denoted $A\; -\; B$, may be defined straightforwardly via the membership function:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\backslash setminus\{B\}\}(x)\; =\; t(\backslash mu\_A(x),n(\backslash mu\_B(x))),$

:which means $A\; \backslash setminus\; B\; =\; A\; \backslash cap\; \backslash neg\{B\}$, e. g.:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\backslash setminus\{B\}\}(x)\; =\; \backslash min(\backslash mu\_A(x),1\; -\; \backslash mu\_B(x)).$N.R. Vemuri, A.S. Hareesh, M.S. Srinath: [http://www.math.sk/fsta2014/presentations/VemuriHareeshSrinath.pdf 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:

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A-\{B\}\}(x)\; =\; \backslash mu\_A(x)\; -\; t(\backslash mu\_A(x),\backslash mu\_B(x)).$

• Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the absolute value, giving

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\; \backslash triangle\; B\}(x)\; =\; |\backslash mu\_A(x)\; -\; \backslash mu\_B(x)|,$

:or by using a combination of just {{math|max}}, {{math|min}}, and standard negation, giving

::$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\; \backslash triangle\; B\}(x)\; =\; \backslash max(\backslash min(\backslash mu\_A(x),\; 1\; -\; \backslash mu\_B(x)),\; \backslash min(\backslash mu\_B(x),\; 1\; -\; \backslash mu\_A(x))).$

: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.

In contrast to the general ambiguity of intersection and union operations, there is clearness for disjoint fuzzy sets:

Two fuzzy sets $A,\; B$ are disjoint iff

:$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; =\; 0\; \backslash lor\; \backslash mu\_B(x)\; =\; 0$

which is equivalent to

:$\backslash nexists$ $x\; \backslash in\; U:\; \backslash mu\_A(x)\; >\; 0\; \backslash land\; \backslash mu\_B(x)\; >\; 0$

and also equivalent to

:$\backslash forall\; x\; \backslash in\; U:\; \backslash min(\backslash mu\_A(x),\backslash mu\_B(x))\; =\; 0$

We keep in mind that {{math|min}}/{{math|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 $A,\; B$ any intersection will give ∅, and any union will give the same result, which is denoted as

:$A\; \backslash ,\backslash dot\{\backslash cup\}\backslash ,\; B\; =\; A\; \backslash cup\; B$

with its membership function given by

:$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{A\; \backslash dot\{\backslash cup\}\; B\}(x)\; =\; \backslash mu\_A(x)\; +\; \backslash mu\_B(x)$

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

For disjoint fuzzy sets $A,\; B$ the following holds true:

:$\backslash operatorname\{Supp\}(A\; \backslash ,\backslash dot\{\backslash cup\}\backslash ,\; B)\; =\; \backslash operatorname\{Supp\}(A)\; \backslash cup\; \backslash operatorname\{Supp\}(B)$

This can be generalized to finite families of fuzzy sets as follows:

Given a family $A\; =\; (A\_i)\_\{i\; \backslash in\; I\}$ of fuzzy sets with index set I (e.g. I = {1,2,3,...,n}). This family is (pairwise) disjoint iff

:$\backslash text\{for\; all\; \}\; x\; \backslash in\; U\; \backslash text\{\; there\; exists\; at\; most\; one\; \}\; i\; \backslash in\; I\; \backslash text\{\; such\; that\; \}\; \backslash mu\_\{A\_i\}(x)\; >\; 0.$

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

:$\backslash dot\{\backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\}\backslash ,\; A\_i\; =\; \backslash bigcup\_\{i\; \backslash in\; I\}\; A\_i$

with its membership function given by

:$\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_\{\backslash dot\{\backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\}\; A\_i\}(x)\; =\; \backslash sum\_\{i\; \backslash in\; I\}\; \backslash mu\_\{A\_i\}(x)$

Again only one of the summands is greater than zero.

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

:$\backslash operatorname\{Supp\}\backslash left(\backslash dot\{\backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\}\backslash ,\; A\_i\backslash right)\; =\; \backslash bigcup\backslash limits\_\{i\; \backslash in\; I\}\; \backslash operatorname\{Supp\}(A\_i)$

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

:$\backslash operatorname\{Card\}(A)\; =\; \backslash operatorname\{sc\}(A)\; =\; |A|\; =\; \backslash sum\_\{x\; \backslash in\; U\}\; \backslash mu\_A(x)$.

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

:$\backslash operatorname\{RelCard\}(A)\; =\; \backslash |A\backslash |\; =\; \backslash operatorname\{sc\}(A)/|U|\; =\; |A|/|U|$.

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

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

which looks very similar to the expression for conditional probability.

Note:

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

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

:$d(\backslash alpha,\backslash beta)\; =\; \backslash |\; \backslash alpha\; -\; \backslash beta\; \backslash |$.

For instance, if $U$ is finite, i.e. $U\; =\; \backslash \{x\_1,\; x\_2,\; ...\; x\_n\backslash \}$, such a metric may be defined by:

:$d(\backslash alpha,\backslash beta)\; :=\; \backslash max\; \backslash \{\; |\backslash alpha(x\_i)\; -\; \backslash beta(x\_i)|\; :\; i=1,\; ...,\; n\; \backslash \}$ where $\backslash alpha$ and $\backslash beta$ are sequences of real numbers between 0 and 1.

For infinite $U$, 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:

:$d(A,B)\; :=\; d(\backslash mu\_A,\backslash mu\_B)$,

which becomes in the above sample:

:$d(A,B)\; =\; \backslash max\; \backslash \{\; |\backslash mu\_A(x\_i)\; -\; \backslash mu\_B(x\_i)|\; :\; i=1,...,n\; \backslash \}$.

Again for infinite $U$ 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., $\backslash varnothing$ and $U$.

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

:$S\; =\; 1\; /\; (1\; +\; d(A,B))$ if $d(A,B)$ is finite, $0$ else,

or after Williams and Steele:

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

where $\backslash alpha\; >\; 0$ is a steepness parameter and $\backslash exp(x)\; =\; e^x$.{{Cn|date=December 2024}}

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 $L$ of a given kind; usually it is required that $L$ 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.{{cite journal | doi=10.1016/0022-247X(67)90189-8 | title=L-fuzzy sets | date=1967 | last1=Goguen | first1=J.A |author-link=Joseph Goguen | journal=Journal of Mathematical Analysis and Applications | volume=18 | pages=145–174 | doi-access=free }} 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) $A$ is characterized by two functions:

:1. $\backslash mu\_A(x)$ – degree of membership of x

:2. $\backslash nu\_A(x)$ – degree of non-membership of x

with functions $\backslash mu\_A,\; \backslash nu\_A:\; U\; \backslash to\; [0,1]$ with $\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; +\; \backslash nu\_A(x)\; \backslash le\; 1$.

This resembles a situation like some person denoted by $x$ voting

• for a proposal $A$: ($\backslash mu\_A(x)=1,\; \backslash nu\_A(x)=0$),
• against it: ($\backslash mu\_A(x)=0,\; \backslash nu\_A(x)=1$),
• or abstain from voting: ($\backslash mu\_A(x)=\backslash nu\_A(x)=0$).

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 $D^*\; =\; \backslash \{(\backslash alpha,\backslash beta)\; \backslash in\; [0,\; 1]^2\; :\; \backslash alpha\; +\; \backslash beta\; =\; 1\; \backslash \}$ and by combining both functions to $(\backslash mu\_A,\backslash nu\_A):\; U\; \backslash to\; D^*$ 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]: $\backslash mu\_A,\; \backslash eta\_A,\; \backslash nu\_A$, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition $\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; +\; \backslash eta\_A(x)\; +\; \backslash nu\_A(x)\; \backslash le\; 1$

This expands the voting sample above by an additional possibility of "refusal of voting".

With $D^*\; =\; \backslash \{(\backslash alpha,\backslash beta,\backslash gamma)\; \backslash in\; [0,\; 1]^3\; :\; \backslash alpha\; +\; \backslash beta\; +\; \backslash gamma\; =\; 1\; \backslash \}$ and special "picture fuzzy" negators, t- and s-norms this resembles just another type of L-fuzzy sets.Bui Cong Cuong, Vladik Kreinovich, Roan Thi Ngan: [http://digitalcommons.utep.edu/cgi/viewcontent.cgi?article=2050&context=cs_techrep A classification of representable t-norm operators for picture fuzzy sets], in: Departmental Technical Reports (CS). Paper 1047, 2016

One extension of IFS is what is known as Pythagorean fuzzy sets. Such sets satisfy the constraint $\backslash mu\_A(x)^2\; +\; \backslash nu\_A(x)^2\; \backslash le\; 1$, which is reminiscent of the Pythagorean theorem.{{Cite book|last=Yager|first=Ronald R. |title=2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS) |chapter=Pythagorean fuzzy subsets |date=June 2013 |pages=57–61|doi=10.1109/IFSA-NAFIPS.2013.6608375|isbn=978-1-4799-0348-1|s2cid=36286152}}{{Cite journal|last=Yager|first=Ronald R|date=2013|title=Pythagorean membership grades in multicriteria decision making|journal=IEEE Transactions on Fuzzy Systems|volume=22|issue=4|pages=958–965|doi=10.1109/TFUZZ.2013.2278989|s2cid=37195356}}{{Cite book|title=Properties and applications of Pythagorean fuzzy sets.|last=Yager|first=Ronald R.|publisher=Springer |location=Cham|date=December 2015|isbn=978-3-319-26302-1|pages=119–136}} Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of $\backslash mu\_A(x)\; +\; \backslash nu\_A(x)\; \backslash le\; 1$ is not valid. However, the less restrictive condition of $\backslash mu\_A(x)^2\; +\; \backslash nu\_A(x)^2\; \backslash le\; 1$ may be suitable in more domains.{{cite journal | vauthors = Yanase J, Triantaphyllou E| title = A Systematic Survey of Computer-Aided Diagnosis in Medicine: Past and Present Developments. | journal = Expert Systems with Applications | volume = 138 | pages = 112821 | date = 2019 | doi = 10.1016/j.eswa.2019.112821 | s2cid = 199019309 }}{{Cite journal|vauthors = Yanase J, Triantaphyllou E|date=2019|title=The Seven Key Challenges for the Future of Computer-Aided Diagnosis in Medicine.|doi=10.1016/j.ijmedinf.2019.06.017|pmid=31445285|journal= International Journal of Medical Informatics|volume=129|pages=413–422|s2cid=198287435 }}

{{main|Fuzzy logic}}

As an extension of the case of multi-valued logic, valuations ($\backslash mu\; :\; \backslash mathit\{V\}\_o\; \backslash to\; \backslash mathit\{W\}$) of propositional variables ($\backslash mathit\{V\}\_o$) into a set of membership degrees ($\backslash mathit\{W\}$) 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.Siegfried Gottwald, 2001. A Treatise on Many-Valued Logics. Baldock, Hertfordshire, England: Research Studies Press Ltd., {{ISBN|978-0-86380-262-1}}

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."{{cite journal | doi=10.1016/0020-0255(75)90036-5 | title=The concept of a linguistic variable and its application to approximate reasoning—I | date=1975 | last1=Zadeh | first1=L.A. | journal=Information Sciences | volume=8 | issue=3 | pages=199–249 }}

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

{{main|Fuzzy number}}

A fuzzy number{{cite journal | doi=10.1016/S0165-0114(99)80004-9 | title=Fuzzy sets as a basis for a theory of possibility | date=1999 | last1=Zadeh | first1=L.A. | journal=Fuzzy Sets and Systems | volume=100 | pages=9–34 }} is a fuzzy set that satisfies all the following conditions:

• A is normalised;
• A is a convex set;
• The membership function $\backslash mu\_\{A\}(x)$ achieves the value 1 at least once;
• The membership function $\backslash mu\_\{A\}(x)$ is at least segmentally continuous.

If these conditions are not satisfied, then A is not a fuzzy number. The core of this fuzzy number is a singleton; its location is:

:: $\backslash ,\; C(A)\; =\; x^*\; :\; \backslash mu\_A(x^*)=1$

Fuzzy numbers can be likened to the funfair game "guess your weight," where someone guesses the contestant's weight, with closer guesses being more correct, and where the guesser "wins" if he or she guesses near enough to the contestant's weight, with the actual weight being completely correct (mapping to 1 by the membership function).

The kernel $K(A)\; =\; \backslash operatorname\{Kern\}(A)$ of a fuzzy interval $A$ 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 $\backslash R$ where $\backslash mu\_A(x)$ 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.

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,J. A. Goguen "Categories of fuzzy sets: applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968 led to the development of Goguen categories in the 21st century.Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations" 2007 Springer {{ISBN|9781402061639}}{{cite journal | doi=10.1016/S0165-0114(02)00508-0 | title=Representation theory of Goguen categories | date=2003 | last1=Winter | first1=Michael | journal=Fuzzy Sets and Systems | volume=138 | pages=85–126 }} In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in L-fuzzy sets.{{cite journal | doi=10.1016/0022-247X(67)90189-8 | title=L-fuzzy sets | date=1967 | last1=Goguen | first1=J.A | journal=Journal of Mathematical Analysis and Applications | volume=18 | pages=145–174 | doi-access=free }}

There are numerous mathematical extensions similar to or more general than fuzzy sets. Since fuzzy sets were introduced in 1965 by Zadeh, many new mathematical constructions and theories treating imprecision, inaccuracy, vagueness, uncertainty and vulnerability have been developed. Some of these constructions and theories are extensions of fuzzy set theory, while others attempt to mathematically model inaccuracy/vagueness and uncertainty in a different way. The diversity of such constructions and corresponding theories includes:

• Fuzzy Sets (Zadeh, 1965)
• interval sets (Moore, 1966),
• L-fuzzy sets (Goguen, 1967),
• flou sets (Gentilhomme, 1968),
• type-2 fuzzy sets and type-n fuzzy sets (Zadeh, 1975),
• interval-valued fuzzy sets (Grattan-Guinness, 1975; Jahn, 1975; Sambuc, 1975; Zadeh, 1975),
• level fuzzy sets (Radecki, 1977)
• 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),
• vague sets (Wen-Lung Gau and Buehrer, 1993),
• α-level sets (Yao, 1997),
• shadowed sets (Pedrycz, 1998),
• neutrosophic sets (NSs) (Smarandache, 1998),
• bipolar fuzzy sets (Wen-Ran Zhang, 1998),
• genuine sets (Demirci, 1999),
• soft sets (Molodtsov, 1999),
• complex fuzzy set (2002),
• intuitionistic fuzzy rough sets (Cornelis, De Cock and Kerre, 2003)
• L-fuzzy rough sets (Radzikowska and Kerre, 2004),
• multi-fuzzy sets (Sabu Sebastian, 2009),
• 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)
• pythagorean fuzzy set (Yager , 2013),
• picture fuzzy set (Cuong, 2013),
• spherical fuzzy set (Mahmood, 2018).

{{More citations needed section|date=November 2015}}

The fuzzy relation equation is an equation of the form {{nowrap|1=A · R = B}}, where A and B are fuzzy sets, R is a fuzzy relation, and {{nowrap|A · R}} stands for the composition of A with R {{Citation needed|date=September 2017}}.

A measure d of fuzziness for fuzzy sets of universe $U$ should fulfill the following conditions for all $x\; \backslash in\; U$:

1. $d(A)\; =\; 0$ if $A$ is a crisp set: $\backslash mu\_A(x)\; \backslash in\; \backslash \{0,\backslash ,1\backslash \}$
2. $d(A)$ has a unique maximum if $\backslash forall\; x\; \backslash in\; U:\; \backslash mu\_A(x)\; =\; 0.5$
3. $\backslash forall\; x\; \backslash in\; U:(\backslash mu\_A(x)\; \backslash leq\; \backslash mu\_B(x)\; \backslash leq\; 0.5)\; \backslash or\; (\backslash mu\_A(x)\; \backslash geq\; \backslash mu\_B(x)\; \backslash geq\; 0.5)$

::$\backslash Rightarrow\; d(A)\; \backslash leq\; d(B)$,

::which means that B is "crisper" than A.

1. $d(\backslash neg\{A\})\; =\; d(A)$

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

For finite $U\; =\; \backslash \{x\_1,\; x\_2,\; ...\; x\_n\backslash \}$ the entropy of a fuzzy set $A$ is given by

:$d(A)\; =\; H(A)\; +\; H(\backslash neg\{A\})$,

::$H(A)\; =\; -k\; \backslash sum\_\{i=1\}^n\; \backslash mu\_A(x\_i)\; \backslash ln\; \backslash mu\_A(x\_i)$

or just

:$d(A)\; =\; -k\; \backslash sum\_\{i=1\}^n\; S(\backslash mu\_A(x\_i))$

where $S(x)\; =\; H\_e(x)$ is Shannon's function (natural entropy function)

:$S(\backslash alpha)\; =\; -\backslash alpha\; \backslash ln\; \backslash alpha\; -\; (1-\backslash alpha)\; \backslash ln\; (1-\backslash alpha),\backslash \; \backslash alpha\; \backslash in\; [0,1]$

and $k$ 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 $A$ be a fuzzy set with a continuous membership function (fuzzy variable). Then

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

and its entropy is

:$d(A)\; =\; -k\; \backslash int\_\{-\; \backslash infty\}^\backslash infty\; S(\backslash operatorname\{Cr\}\; \backslash lbrace\; A\; \backslash geq\; t\; \backslash rbrace\; )\backslash ,dt.${{cite journal|doi=10.1016/0165-0114(92)90239-Z|title=Entropy, distance measure and similarity measure of fuzzy sets and their relations|journal=Fuzzy Sets and Systems|volume=52|issue=3|pages=305–318|year=1992|last1=Xuecheng|first1=Liu}}{{cite journal|doi=10.1186/s40467-015-0029-5|title=Fuzzy cross-entropy|journal=Journal of Uncertainty Analysis and Applications|volume=3|year=2015|last1=Li|first1=Xiang|doi-access=free}}

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.

{{harvnb|Burgin|Chunihin|1997}}; {{harvnb|Kerre|2001}}; {{harvnb|Deschrijver|Kerre|2003}}.

{{div col|colwidth=25em}}

• 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

{{div col end}}

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{{refend}}

{{Non-classical logic}}

{{Set theory}}

{{DEFAULTSORT:Fuzzy Set}}

Category:Fuzzy logic

Category:Systems of set theory