You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 3.83.0.0/16 has been blocked (disabled) by ‪ST47‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider. To prevent abuse, web hosts may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a web host's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: ((unblock|reason=Caught by a web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~)). You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and they should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 21:48, 22 February 2026.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
Your IP address is in a range that has been blocked on all Wikimedia Foundation wikis. The block was made by ‪AntiCompositeNumber‬. The reason given is Open proxy/Webhost: Visit the FAQ if you are affected .
- Start of block: 15:08, 18 June 2022
- Expiry of block: 15:08, 18 April 2025
Your current IP address is 3.83.16.50. The blocked range is 3.83.0.0/16. Please include all above details in any queries you make. If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

You can view and copy the source of this page:

((Short description|Sets whose elements have degrees of membership))
In [[mathematics]], '''fuzzy sets''' (a.k.a. '''uncertain sets''') are [[Set (mathematics)|sets]] whose [[Element (mathematics)|elements]] have degrees of membership. Fuzzy sets were introduced independently by [[Lotfi Asker Zadeh|Lotfi A. Zadeh]] in 1965 as an extension of the classical notion of set.<ref>L. A. Zadeh (1965) [http://www.cs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-Information%20and%20Control-1965.pdf "Fuzzy sets"] ((Webarchive|url=https://web.archive.org/web/20150813153834/http://www.cs.berkeley.edu/~zadeh/papers/Fuzzy%20Sets-Information%20and%20Control-1965.pdf |date=2015-08-13 )). ''Information and Control'' 8 (3) 338–353.</ref><ref>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 ((Cite journal | last1 = Gottwald | first1 = S. | title = An early approach toward graded identity and graded membership in set theory | doi = 10.1016/j.fss.2009.12.005 | journal = Fuzzy Sets and Systems | volume = 161 | issue = 18 | pages = 2369–2379 | year = 2010 ))</ref>
At the same time, ((harvtxt|Salii|1965)) defined a more general kind of structure called an [[L-relation|''L''-relation]], which he studied in an [[abstract algebra]]ic context. Fuzzy relations, which are now used throughout [[fuzzy mathematics]] and have applications in areas such as [[linguistics]] ((harv|De Cock|Bodenhofer|Kerre|2000)), [[Decision making|decision-making]] ((harv|Kuzmin|1982)), and [[Cluster analysis|clustering]] ((harv|Bezdek|1978)), are special cases of ''L''-relations when ''L'' is the [[unit interval]] [0,&thinsp;1].

In classical [[set theory]], the membership of elements in a set is assessed in binary terms according to a [[Principle of bivalence|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 (mathematics)|membership function]] valued in the [[Real number|real]] unit interval [0,&thinsp;1]. Fuzzy sets generalize classical sets, since the [[indicator function]]s (aka characteristic functions) of classical sets are special cases of the membership functions of fuzzy sets, if the latter only takes values 0 or 1.<ref name=":0">D. Dubois and H. Prade (1988) Fuzzy Sets and Systems. Academic Press, New York.</ref> 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]].<ref>((Cite journal | doi=10.1186/1471-2105-7-S4-S7| pmid=17217525| pmc=1780132| title=FM-test: A fuzzy-set-theory-based approach to differential gene expression data analysis| journal=BMC Bioinformatics| volume=7| pages=S7| year=2006| last1=Liang| first1=Lily R.| last2=Lu| first2=Shiyong| last3=Wang| first3=Xuena| last4=Lu| first4=Yi| last5=Mandal| first5=Vinay| last6=Patacsil| first6=Dorrelyn| last7=Kumar| first7=Deepak| issue=Suppl 4| doi-access=free))</ref>

==Definition==

A fuzzy set is a pair <math>(U, m)</math> where <math>U</math> is a set (often required to be [[empty set|non-empty]]) and <math>m\colon U \rightarrow [0,1]</math> a membership function. 
The reference set <math>U</math> (sometimes denoted by <math>\Omega</math> or <math>X</math>) is called '''universe of discourse''', and for each <math>x\in U,</math> the value <math>m(x)</math> is called the '''grade''' of membership of <math>x</math> in <math>(U,m)</math>. 
The function <math>m = \mu_A</math> is called the '''membership function''' of the fuzzy set <math>A = (U, m)</math>.

For a finite set <math>U=\{x_1,\dots,x_n\},</math> the fuzzy set <math>(U, m)</math> is often denoted by <math>\{m(x_1)/x_1,\dots,m(x_n)/x_n\}.</math>

Let <math>x \in U</math>. Then <math>x</math> is called 
* '''not included''' in the fuzzy set <math>(U,m)</math> if ((nowrap|<math>m(x) = 0</math>)) (no member), 
* '''fully included''' if ((nowrap|<math>m(x) = 1</math>)) (full member), 
* '''partially included''' if ((nowrap|<math>0 < m(x) < 1</math> (fuzzy member).<ref>((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))</ref>))
The (crisp) set of all fuzzy sets on a universe <math>U</math> is denoted with <math>SF(U)</math> (or sometimes just <math>F(U)</math>).<ref name="Beg20019" />

===Crisp sets related to a fuzzy set===
For any fuzzy set <math>A = (U,m)</math> and <math>\alpha \in [0,1]</math> the following crisp sets are defined:
* <math>A^{\ge\alpha} = A_\alpha = \{x \in U \mid m(x)\ge\alpha\}</math> is called its '''α-cut''' (aka '''α-level set''') 
* <math>A^{>\alpha} = A'_\alpha = \{x \in U \mid m(x)>\alpha\}</math> is called its '''strong α-cut''' (aka '''strong α-level set''') 
* <math>S(A) = \operatorname{Supp}(A) = A^{>0} = \{x \in U \mid m(x)>0\}</math> is called its '''support''' 
* <math>C(A) = \operatorname{Core}(A) = A^{=1} = \{x \in U \mid m(x)=1\}</math> is called its '''core''' (or sometimes '''kernel''' <math>\operatorname{Kern}(A)</math>).

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

===Other definitions===
* A fuzzy set <math>A = (U,m)</math> is '''empty''' (<math>A = \varnothing</math>) [[iff]] (if and only if)
::[[Universal quantification#notation|<math>\forall</math>]]<math> x \in U: \mu_A(x) = m(x) = 0</math>

* Two fuzzy sets <math>A</math> and <math>B</math> are '''equal''' (<math>A = B</math>) iff
::<math>\forall x \in U: \mu_A(x) = \mu_B(x)</math>

* A fuzzy set <math>A</math> is '''included''' in a fuzzy set <math>B</math> (<math>A \subseteq B</math>) iff
::<math>\forall x \in U: \mu_A(x) \le \mu_B(x)</math>

* For any fuzzy set <math>A</math>, any element <math>x \in U</math> that satisfies
::<math>\mu_A(x) = 0.5</math> 
:is called a '''crossover point'''.

* Given a fuzzy set <math>A</math>, any <math>\alpha \in [0,1]</math>, for which <math>A^{=\alpha} = \{x \in U \mid \mu_A(x) = \alpha\}</math> is not empty, is called a '''level''' of A. 
* The '''level set''' of A is the set of all levels <math>\alpha\in[0,1]</math> representing distinct cuts. It is the [[image (mathematics)|image]] of <math>\mu_A</math>:
::<math>\Lambda_A = \{\alpha \in [0,1] : A^{=\alpha} \ne \varnothing\} = \{\alpha \in [0, 1] : {}</math>[[Existential quantification|<math>\exist</math>]]<math>x \in U(\mu_A(x) = \alpha)\} = \mu_A(U)</math>

* For a fuzzy set <math>A</math>, its '''height''' is given by
::<math>\operatorname{Hgt}(A) = \sup \{\mu_A(x) \mid x \in U\} = \sup(\mu_A(U))</math>
:where <math>\sup</math> denotes the [[Infimum and supremum|supremum]], which exists because <math>\mu_A(U)</math> is non-empty and bounded above by 1. If ''U'' is finite, we can simply replace the supremum by the maximum.

* A fuzzy set <math>A</math> is said to be '''normalized''' iff
::<math>\operatorname{Hgt}(A) = 1</math>
: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 <math>A</math> may be normalized with result <math>\tilde{A}</math> by dividing the membership function of the fuzzy set by its height:
::<math>\forall x \in U: \mu_{\tilde{A))(x) = \mu_A(x)/\operatorname{Hgt}(A)</math>
:Besides similarities this differs from the usual [[normalizing constant|normalization]] in that the normalizing constant is not a sum.

* For fuzzy sets <math>A</math> of real numbers (''U'' ⊆ ℝ) with [[bounded set|bounded]] support, the '''width''' is defined as
::<math>\operatorname{Width}(A) = \sup(\operatorname{Supp}(A)) - \inf(\operatorname{Supp}(A))</math>
:In the case when <math>\operatorname{Supp}(A)</math> is a finite set, or more generally a [[closed set]], the width is just
::<math>\operatorname{Width}(A) = \max(\operatorname{Supp}(A)) - \min(\operatorname{Supp}(A))</math>
:In the ''n''-dimensional case (''U'' ⊆ ℝ<sup>''n''</sup>) the above can be replaced by the ''n''-dimensional volume of <math>\operatorname{Supp}(A)</math>. 
:In general, this can be defined given any [[Measure (mathematics)|measure]] on ''U'', for instance by integration (e.g. [[Lebesgue integration]]) of <math>\operatorname{Supp}(A)</math>.

* A real fuzzy set <math>A</math> (''U'' ⊆ ℝ) is said to be '''convex''' (in the fuzzy sense, not to be confused with a crisp [[convex set]]), iff
::<math>\forall x,y \in U, \forall\lambda\in[0,1]: \mu_A(\lambda{x} + (1-\lambda)y) \ge \min(\mu_A(x),\mu_A(y))</math>.
: Without loss of generality, we may take ''x'' ≤ ''y'', which gives the equivalent formulation
::<math>\forall z \in [x,y]: \mu_A(z) \ge \min(\mu_A(x),\mu_A(y))</math>.
: This definition can be extended to one for a general [[topological space]] ''U'': we say the fuzzy set <math>A</math> is '''convex''' when, for any subset ''Z'' of ''U'', the condition  
::<math>\forall z \in Z: \mu_A(z) \ge \inf(\mu_A(\partial Z))</math>
: holds, where <math>\partial Z</math> denotes the [[Boundary (topology)|boundary]] of ''Z'' and <math>f(X) = \{f(x) \mid x \in X\}</math> denotes the [[image (mathematics)|image]] of a set ''X'' (here <math>\partial Z</math>) under a function ''f'' (here <math>\mu_A</math>).

===Fuzzy set operations===
((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 <math>A</math>, its '''complement''' <math>\neg{A}</math> (sometimes denoted as <math>A^c</math> or <math>cA</math>) is defined by the following membership function:
::<math>\forall x \in U: \mu_{\neg{A))(x) = 1 - \mu_A(x)</math>.
* Let t be a [[t-norm]], and s the corresponding s-norm (aka t-conorm). Given a pair of fuzzy sets <math>A, B</math>, their '''intersection''' <math>A\cap{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cap{B))(x) = t(\mu_A(x),\mu_B(x))</math>,
:and their '''union''' <math>A\cup{B}</math> is defined by:
::<math>\forall x \in U: \mu_{A\cup{B))(x) = s(\mu_A(x),\mu_B(x))</math>.

By the definition of the t-norm, we see that the union and intersection are [[commutative]], [[monotonic]], [[associative]], and have both a [[absorbing element|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 [[Indexed family|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:
* <math>\forall x \in U: \mu_{A\cup{B))(x) = \max(\mu_A(x),\mu_B(x))</math> and <math>\mu_{A\cap{B))(x) = \min(\mu_A(x),\mu_B(x))</math>.<ref name="BGFuzzy">((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))</ref>

* If the standard negator <math>n(\alpha) = 1 - \alpha, \alpha \in [0, 1]</math> is replaced by another [[t-norm#Non-standard negators|strong negator]], the fuzzy set difference may be generalized by
::<math>\forall x \in U: \mu_{\neg{A))(x) = n(\mu_A(x)).</math>
* 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-norm]]s.

:The fuzzy intersection is not [[Idempotence|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 <math>A</math> and <math>\nu \in \R^+</math> the &nu;-th power of <math>A</math> is defined by the membership function:
::<math>\forall x \in U: \mu_{A^{\nu))(x) = \mu_{A}(x)^{\nu}.</math>
The case of exponent two is special enough to be given a name.
* For any fuzzy set <math>A</math> the '''concentration''' <math>CON(A) = A^2</math> is defined
::<math>\forall x \in U: \mu_{CON(A)}(x) = \mu_{A^2}(x) = \mu_{A}(x)^2.</math>
Taking <math>0^0 = 1</math>, we have <math>A^0 = U</math> and <math>A^1 = A.</math>

* Given fuzzy sets <math>A, B</math>, the fuzzy set '''difference''' <math>A \setminus B</math>, also denoted <math> A - B</math>, may be defined straightforwardly via the membership function:
::<math>\forall x \in U: \mu_{A\setminus{B))(x) = t(\mu_A(x),n(\mu_B(x))),</math>
:which means <math>A \setminus B = A \cap \neg{B}</math>, e. g.:
::<math>\forall x \in U: \mu_{A\setminus{B))(x) = \min(\mu_A(x),1 - \mu_B(x)).</math><ref name="Vemuri2014">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</ref>

:Another proposal for a set difference could be: 
::<math>\forall x \in U: \mu_{A-{B))(x) = \mu_A(x) - t(\mu_A(x),\mu_B(x)).</math><ref name="Vemuri2014" />

* Proposals for symmetric fuzzy set differences have been made by Dubois and Prade (1980), either by taking the [[absolute value]], giving
::<math>\forall x \in U: \mu_{A \triangle B}(x) = |\mu_A(x) - \mu_B(x)|,</math>
:or by using a combination of just ((math|max)), ((math|min)), and standard negation, giving
::<math>\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))).</math><ref name="Vemuri2014" />

: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).<ref name="Vemuri2014" />

* 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 <math>A, B</math> are '''disjoint''' iff
:<math>\forall x \in U: \mu_A(x) = 0 \lor \mu_B(x) = 0</math>
which is equivalent to
:[[Existential quantification#Negation|<math>\nexists</math>]] <math>x \in U: \mu_A(x) > 0 \land \mu_B(x) > 0</math>
and also equivalent to 
:<math>\forall x \in U: \min(\mu_A(x),\mu_B(x)) = 0</math>
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 sets|disjoint]] according to the standard definition for crisp sets.

For disjoint fuzzy sets <math>A, B</math> any intersection will give ∅, and any union will give the same result, which is denoted as
:<math>A \,\dot{\cup}\, B = A \cup B</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{A \dot{\cup} B}(x) = \mu_A(x) + \mu_B(x)</math>
Note that only one of both summands is greater than zero.

For disjoint fuzzy sets <math>A, B</math> the following holds true:
:<math>\operatorname{Supp}(A \,\dot{\cup}\, B) = \operatorname{Supp}(A) \cup \operatorname{Supp}(B)</math>

This can be generalized to finite families of fuzzy sets as follows:
Given a family <math>A = (A_i)_{i \in I}</math> of fuzzy sets with index set ''I'' (e.g. ''I'' = {1,2,3,...,''n''}). This family is '''(pairwise) disjoint''' iff
:<math>\text{for all } x \in U \text{ there exists at most one } i \in I \text{ such that } \mu_{A_i}(x) > 0.</math>

A family of fuzzy sets <math>A = (A_i)_{i \in I}</math> is disjoint, iff the family of underlying supports <math>\operatorname{Supp} \circ A = (\operatorname{Supp}(A_i))_{i \in I}</math> 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:
:<math>\dot{\bigcup\limits_{i \in I))\, A_i = \bigcup_{i \in I} A_i</math> 
with its membership function given by
:<math>\forall x \in U: \mu_{\dot{\bigcup\limits_{i \in I)) A_i}(x) = \sum_{i \in I} \mu_{A_i}(x)</math>
Again only one of the summands is greater than zero.

For disjoint families of fuzzy sets <math>A = (A_i)_{i \in I}</math> the following holds true:
:<math>\operatorname{Supp}\left(\dot{\bigcup\limits_{i \in I))\, A_i\right) = \bigcup\limits_{i \in I} \operatorname{Supp}(A_i)</math>

===Scalar cardinality===
For a fuzzy set <math>A</math> with finite support <math>\operatorname{Supp}(A)</math> (i.e. a "finite fuzzy set"), its '''cardinality''' (aka '''scalar cardinality''' or '''sigma-count''') is given by
:<math>\operatorname{Card}(A) = \operatorname{sc}(A) = |A| = \sum_{x \in U} \mu_A(x)</math>.
In the case that ''U'' itself is a finite set, the '''relative cardinality''' is given by 
:<math>\operatorname{RelCard}(A) = \|A\| = \operatorname{sc}(A)/|U| = |A|/|U|</math>.
This can be generalized for the divisor to be a non-empty fuzzy set: For fuzzy sets <math>A,G</math> with ''G'' ≠ ∅, we can define the '''relative cardinality''' by:
:<math>\operatorname{RelCard}(A,G) = \operatorname{sc}(A|G) = \operatorname{sc}(A\cap{G})/\operatorname{sc}(G)</math>,
which looks very similar to the expression for [[conditional probability]]<!--and because of that, sc(A|G) is used instead of sc(A/G)- this isn't **set** division, isn't it? -->.
Note:
* <math>\operatorname{sc}(G) > 0</math> here.
* The result may depend on the specific intersection (t-norm) chosen. 
* For <math>G = U</math> the result is unambiguous and resembles the prior definition.

===Distance and similarity===
For any fuzzy set <math>A</math> the membership function <math>\mu_A: U \to [0,1]</math> can be regarded as a family <math>\mu_A = (\mu_A(x))_{x \in U} \in [0,1]^U</math>. The latter is a [[metric space]] with several metrics <math>d</math> known. A metric can be derived from a [[Norm (mathematics)|norm]] (vector norm) <math>\|\,\|</math> via
:<math>d(\alpha,\beta) = \| \alpha - \beta \|</math>.
For instance, if <math>U</math> is finite, i.e. <math>U = \{x_1, x_2, ... x_n\}</math>, such a metric may be defined by: 
:<math>d(\alpha,\beta) := \max \{ |\alpha(x_i) - \beta(x_i)| : i=1, ..., n \}</math> where <math>\alpha</math> and <math>\beta</math> are sequences of real numbers between 0 and 1. 
For infinite <math>U</math>, 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:
:<math>d(A,B) := d(\mu_A,\mu_B)</math>,
which becomes in the above sample:
:<math>d(A,B) = \max \{ |\mu_A(x_i) - \mu_B(x_i)| : i=1,...,n \}</math>.
Again for infinite <math>U</math> 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., <math>\varnothing</math> and <math>U</math>.

Similarity measures (here denoted by <math>S</math>) may then be derived from the distance, e.g. after a proposal by Koczy:
:<math>S = 1 / (1 + d(A,B))</math> if <math>d(A,B)</math> is finite, <math>0</math> else, 
or after Williams and Steele:
:<math>S = \exp(-\alpha{d(A,B)})</math> if <math>d(A,B)</math> is finite, <math>0</math> else
where <math>\alpha > 0</math> is a steepness parameter and <math>\exp(x) = e^x</math>.<ref name="Beg20019">Ismat Beg, Samina Ashraf: [https://www.researchgate.net/publication/228744370_Similarity_measures_for_fuzzy_sets Similarity measures for fuzzy sets], at: Applied and Computational Mathematics, March 2009, available on Research Gate since November 23rd, 2016</ref>

Another definition for interval valued (rather 'fuzzy') similarity measures <math>\zeta</math> is provided by Beg and Ashraf as well.<ref name="Beg20019" />

===''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) [[algebraic structure|algebra]] or [[structure (mathematical logic)|structure]] <math>L</math> of a given kind; usually it is required that <math>L</math> be at least a [[poset]] or [[lattice (order)|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,&thinsp;1] are then called [0,&thinsp;1]-valued membership functions. These kinds of generalizations were first considered in 1967 by [[Joseph Goguen]], who was a student of Zadeh.<ref>[[Joseph Goguen|Goguen, Joseph A.]], 196, "''L''-fuzzy sets". ''Journal of Mathematical Analysis and Applications'' '''18''': 145–174</ref> A classical corollary may be indicating truth and membership values by {f,&thinsp;t} instead of {0,&thinsp;1}.

An extension of fuzzy sets has been provided by [[Krassimir Atanassov|Atanassov]]. An '''intuitionistic fuzzy set''' (IFS) <math>A</math> is characterized by two functions:
:1. <math>\mu_A(x)</math> – degree of membership of ''x''
:2. <math>\nu_A(x)</math> – degree of non-membership of ''x''
with functions <math>\mu_A, \nu_A: U \to [0,1]</math> with <math>\forall x \in U: \mu_A(x) + \nu_A(x) \le 1</math>.

This resembles a situation like some person denoted by <math>x</math> voting 
* for a proposal <math>A</math>: (<math>\mu_A(x)=1, \nu_A(x)=0</math>), 
* against it: (<math>\mu_A(x)=0, \nu_A(x)=1</math>), 
* or abstain from voting: (<math>\mu_A(x)=\nu_A(x)=0</math>).
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 <math>D^* = \{(\alpha,\beta) \in [0, 1]^2 : \alpha + \beta = 1 \}</math> and by combining both functions to <math>(\mu_A,\nu_A): U \to D^*</math> 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,&thinsp;1]: <math>\mu_A, \eta_A, \nu_A</math>, "degree of positive membership", "degree of neutral membership", and "degree of negative membership" respectively and additional condition <math>\forall x \in U: \mu_A(x) + \eta_A(x) + \nu_A(x) \le 1</math>
This expands the voting sample above by an additional possibility of "refusal of voting".

With <math>D^* = \{(\alpha,\beta,\gamma) \in [0, 1]^3 : \alpha + \beta + \gamma = 1 \}</math> and special "picture fuzzy" negators, t- and s-norms this resembles just another type of ''L''-fuzzy sets.<ref>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</ref><ref>Tridiv Jyoti Neog, Dusmanta Kumar Sut: [https://web.archive.org/web/20171201034358/https://pdfs.semanticscholar.org/dc40/1b8482c784e92fc4c92430510801de59ffab.pdf Complement of an Extended Fuzzy Set], in: International Journal of Computer Applications (097
5–8887), Volume 29 No.3, September 2011</ref>

===Neutrosophic fuzzy sets===
[[File:FuzzyLogic development.png|thumb|Some Key Developments in the Introduction of Fuzzy Set Concepts.<ref name="CADsurvey"/>]]
The concept of IFS has been extended into two major models. The two extensions of IFS are neutrosophic fuzzy sets and Pythagorean fuzzy sets.<ref name="CADsurvey">((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 ))</ref>

Neutrosophic fuzzy sets were introduced by Smarandache in 1998.<ref>((Cite book|title=Neutrosophy: Neutrosophic Probability, Set, and Logic: Analytic Synthesis & Synthetic Analysis.|last=Smarandache|first=Florentin|publisher=American Research Press|year=1998|isbn=978-1879585638))</ref> Like IFS, neutrosophic fuzzy sets have the previous two functions: one for membership <math>\mu_A(x)</math> and another for non-membership <math>\nu_A(x)</math>. The major difference is that neutrosophic fuzzy sets have one more function: for indeterminate <math>i_A(x)</math>. This value indicates that the degree of undecidedness that the entity x belongs to the set. This concept of having indeterminate <math>i_A(x)</math> value can be particularly useful when one cannot be very confident on the membership or non-membership values for item ''x''.<ref name="SevenChallenges">((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 ))</ref> In summary, neutrosophic fuzzy sets are associated with the following functions:

:1. <math>\mu_A(x)</math>—degree of membership of ''x''
:2. <math>\nu_A(x)</math>—degree of non-membership of ''x''
:3. <math>i_A(x)</math>—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 <math>\mu_A(x) + \nu_A(x) \le 1</math>, 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 <math>\mu_A(x)^2 + \nu_A(x)^2 \le 1</math>, which is reminiscent of the Pythagorean theorem.<ref>((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|journal=2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS). IEEE|pages=57–61|doi=10.1109/IFSA-NAFIPS.2013.6608375|isbn=978-1-4799-0348-1|s2cid=36286152))</ref><ref>((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))</ref><ref>((Cite book|title=Properties and applications of Pythagorean fuzzy sets.|last=Yager|first=Ronald R.|publisher=Springer, Cham.|date=December 2015|isbn=978-3-319-26302-1|pages=119–136))</ref> Pythagorean fuzzy sets can be applicable to real life applications in which the previous condition of <math>\mu_A(x) + \nu_A(x) \le 1</math> is not valid. However, the less restrictive condition of <math>\mu_A(x)^2 + \nu_A(x)^2 \le 1</math> may be suitable in more domains.<ref name="CADsurvey"/><ref name="SevenChallenges"/>

== Fuzzy logic ==
((main|Fuzzy logic))

As an extension of the case of [[multi-valued logic]], valuations (<math>\mu : \mathit{V}_o \to \mathit{W}</math>) of [[propositional variable]]s (<math>\mathit{V}_o</math>) into a set of membership degrees (<math>\mathit{W}</math>) can be thought of as [[membership function (mathematics)|membership functions]] mapping [[Predicate (mathematical logic)|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 [[premise]]s from which graded conclusions may be drawn.<ref>[[Siegfried Gottwald]], 2001. ''A Treatise on Many-Valued Logics''. Baldock, Hertfordshire, England: Research Studies Press Ltd., ((ISBN|978-0-86380-262-1))</ref>

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 [[automation|automated]] control and [[knowledge engineering]], and which encompasses many topics involving fuzzy sets and "approximated reasoning."<ref>"[https://pdfs.semanticscholar.org/74d5/e9445c8154c8a77cc71487a4c5cfd9a9588c.pdf The concept of a linguistic variable and its application to approximate reasoning]," ''Information Sciences'' '''8''': 199–249, 301–357; '''9''': 43–80.</ref>

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 ==
((main|Fuzzy number))
A '''fuzzy number'''<ref name="semanticscholar.org">"[https://www.semanticscholar.org/paper/Fuzzy-sets-and-systems%3A-theory-and-applications-Dubois-Prade/a732dce89c9683bd1fdf90ea31acfeadd0a06478  Fuzzy sets as a basis for a theory of possibility]," ''Fuzzy Sets and Systems''</ref> is a fuzzy set that satisfies all the following conditions:
* A is normalised;
* A is a convex set;
* <math>\exists ! x^* \in A, \mu_{A}(x^*)=1</math>;
* The membership function <math>\mu_{A}(x)</math> 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 (mathematics)|singleton]]; its location is:
:: <math> \, C(A) = x^* : \mu_A(x^*)=1</math>

When the condition about the uniqueness of <math>{x^*}</math> is not fulfilled, then the fuzzy set is characterised as a '''fuzzy interval'''.<ref name="semanticscholar.org"/> The core of this fuzzy interval is a crisp interval with:

::<math>\,C(A) = \left[\min\{x \in \R \mid \mu_A(x)=1\} ; \max\{x \in \R \mid \mu_A(x)=1\} \right]</math>.

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 <math>K(A) = \operatorname{Kern}(A)</math> of a fuzzy interval <math>A</math> 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 <math>\R</math> where <math>\mu_A(x)</math> 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<!--'Goguen category' rediretcs here-->==
The use of [[Set theory|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,<ref>J. A. Goguen "Categories of fuzzy sets: applications of non-Cantorian set theory" PhD Thesis University of California, Berkeley, 1968</ref> led to the development of '''Goguen categories'''<!--boldface per WP:R#PLA--> in the 21st century.<ref>Michael Winter "Goguen Categories:A Categorical Approach to L-fuzzy Relations"  2007 [[Springer Verlag|Springer]] ((ISBN|9781402061639))</ref><ref name=goguencateg>Michael Winter "Representation theory of Goguen categories" [[Fuzzy Sets and Systems]] Volume 138, Issue 1, 16 August 2003, Pages 85–126</ref> In these categories, rather than using two valued set membership, more general intervals are used, and may be lattices as in ''L''-fuzzy sets.<ref name=goguencateg/><ref>Goguen, J.A., "L-fuzzy sets". Journal of Mathematical Analysis and Applications 18(1):145–174, 1967</ref>

== Fuzzy relation equation ==
((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 [[Function composition|composition]] of ''A'' with&nbsp;''R'' ((Citation needed|date=September 2017)).

==Entropy==
A measure ''d'' of fuzziness for fuzzy sets of universe <math>U</math> should fulfill the following conditions for all <math>x \in U</math>:
#<math>d(A) = 0</math> if <math>A</math> is a crisp set: <math>\mu_A(x) \in \{0,\,1\}</math>
#<math>d(A)</math> has a unique maximum iff <math>\forall x \in U: \mu_A(x) = 0.5</math>
#<math>\mu_A \leq \mu_B \iff</math>
:::<math>\mu_A \leq  \mu_B \leq 0.5</math>
:::<math>\mu_A \geq \mu_B \geq 0.5</math>
::which means that ''B'' is "crisper" than ''A''.
#<math>d(\neg{A}) = d(A)</math>   
In this case <math>d(A)</math> is called the '''entropy''' of the fuzzy set ''A''.

For '''finite''' <math>U = \{x_1, x_2, ... x_n\}</math> the entropy of a fuzzy set <math>A</math> is given by
:<math>d(A) = H(A) + H(\neg{A})</math>,
::<math>H(A) = -k \sum_{i=1}^n \mu_A(x_i) \ln \mu_A(x_i)</math>
or just
:<math>d(A) = -k \sum_{i=1}^n S(\mu_A(x_i))</math>
where <math>S(x) = H_e(x)</math> is [[Binary entropy function|Shannon's function]] (natural entropy function)
:<math>S(\alpha) = -\alpha \ln \alpha - (1-\alpha) \ln (1-\alpha),\ \alpha \in [0,1]</math>
and <math>k</math> is a constant depending on the measure unit and the logarithm base used (here we have used the natural base [[e (mathematical constant)|e]]).
The physical interpretation of ''k'' is the [[Boltzmann constant]] ''k''<sup>''B''</sup>.

Let <math>A</math> be a fuzzy set with a '''continuous''' membership function (fuzzy variable). Then 
:<math>H(A) = -k \int_{- \infty}^\infty \operatorname{Cr} \lbrace A \geq t \rbrace \ln \operatorname{Cr} \lbrace A \geq t \rbrace \,dt</math>
and its entropy is 
:<math>d(A) = -k \int_{- \infty}^\infty S(\operatorname{Cr} \lbrace A \geq t \rbrace )\,dt.</math><ref>((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))</ref><ref>((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))</ref>

==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.<ref>
((harvnb|Burgin|Chunihin|1997)); ((harvnb|Kerre|2001)); ((harvnb|Deschrijver|Kerre|2003)).</ref>

==See also==
((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))

== References ==
((reflist))

((refbegin|))

== Bibliography ==
* Alkhazaleh, S. and Salleh, A.R. [https://www.hindawi.com/journals/aaa/2012/350603/ Fuzzy Soft Multiset Theory], Abstract and Applied Analysis, 2012, article ID 350600, 20 p.
* [[Krassimir Atanassov|Atanassov, K. T.]] (1983) [http://www.biomed.bas.bg/bioautomation/2016/vol_20.s1/files/20.s1_02.pdf Intuitionistic fuzzy sets], VII ITKR's Session, Sofia (deposited in Central Sci.-Technical Library of Bulg. Acad. of Sci., 1697/84) (in Bulgarian) 
* ((cite journal |doi=10.1016/S0165-0114(86)80034-3 |title=Intuitionistic fuzzy sets |date=1986 |last1=Atanassov |first1=Krassimir T. |journal=Fuzzy Sets and Systems |volume=20 |pages=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.
* ((cite journal|last=Bezdek|first=J.C.|date=1978|title=Fuzzy partitions and relations and axiomatic basis for clustering|journal=Fuzzy Sets and Systems|volume=1|issue=2|pages=111–127|doi=10.1016/0165-0114(78)90012-X))
* ((cite journal |doi=10.1016/0165-0114(89)90218-2 |title=Real-valued multisets and fuzzy sets |date=1989 |last1=Blizard |first1=Wayne D. |journal=Fuzzy Sets and Systems |volume=33 |pages=77–97 )) 
* ((cite journal |doi=10.1016/S0019-9958(71)90288-9 |title=A note on fuzzy sets |date=1971 |last1=Brown |first1=Joseph G. |journal=Information and Control |volume=18 |pages=32–39 ))
* Brutoczki Kornelia: [http://mazsola.iit.uni-miskolc.hu/DATA/diploma/brutoczki_kornelia/fu_gz_01.html Fuzzy Logic] (Diploma) – <small>Although this script has a lot of oddities and intracies due to its incompleteness, it may be used a template for exercise in removing these issues.</small>
* Burgin, M. Theory of Named Sets as a Foundational Basis for Mathematics, in Structures in Mathematical Theories, San Sebastian, 1990, pp.&nbsp; 417–420
* ((cite journal |last1=Burgin |first1=M. |last2=Chunihin |first2=A. |date=1997 |title=Named Sets in the Analysis of Uncertainty |journal=Methodological and Theoretical Problems of Mathematics and Information Sciences |location=Kiev |pages=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.&nbsp;77–84, 2002. ((doi|10.1007/3-540-45813-1_10))
* Chamorro-Martínez, J. et al.: [https://ai2-s2-pdfs.s3.amazonaws.com/0067/203adc1ee4abe030350053722013dbfad8be.pdf 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.&nbsp;619–634
* Chapin, E.W. (1975) Set-valued Set Theory, II, Notre Dame J. Formal Logic, v. 16, pp.&nbsp;255–267
* ((cite journal |doi=10.1111/1468-0394.00250 |title=Intuitionistic fuzzy rough sets: At the crossroads of imperfect knowledge |date=2003 |last1=Cornelis |first1=Chris |last2=De Cock |first2=Martine |last3=Kerre |first3=Etienne E. |journal=Expert Systems |volume=20 |issue=5 |pages=260–270 |s2cid=15031773 ))
* 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.&nbsp;55–95
* ((cite conference|first1=Martine|last1=De Cock|first2=Ulrich|last2=Bodenhofer|first3=Etienne E.|last3=Kerre|title=Modelling Linguistic Expressions Using Fuzzy Relations|date=1–4 October 2000|conference=Proceedings of the 6th International Conference on Soft Computing|location=Iizuka, Japan|pages=353–360|citeseerx=10.1.1.32.8117))
* ((cite journal |doi=10.1016/S0165-0114(97)00235-2 |title=Genuine sets |date=1999 |last1=Demirci |first1=Mustafa |journal=Fuzzy Sets and Systems |volume=105 |issue=3 |pages=377–384 ))
* ((cite journal|last1=Deschrijver|first1=G.|last2=Kerre|first2=E.E.|title=On the relationship between some extensions of fuzzy set theory|journal=Fuzzy Sets and Systems|volume=133|issue=2|pages=227–235|date=2003|doi=10.1016/S0165-0114(02)00127-6))
* ((cite book|editor=Didier Dubois, Henri M. Prade|title=Fundamentals of fuzzy sets|year=2000|publisher=Springer|isbn=978-0-7923-7732-0|series=The Handbooks of Fuzzy Sets Series|volume=7))
* Feng F. [https://ieeexplore.ieee.org/abstract/document/5072885/ 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.&nbsp;47–63 
* Gogen, J.A. (1967) L-fuzzy Sets, Journal Math. Analysis Appl., v. 18, pp.&nbsp;145–174
* ((Cite journal|last1=Gottwald|first1=S.|title=Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part I: Model-Based and Axiomatic Approaches|doi=10.1007/s11225-006-7197-8|journal=Studia Logica|volume=82|issue=2|pages=211–244|year=2006|s2cid=11931230)). ((Cite journal|last1=Gottwald|first1=S.|doi=10.1007/s11225-006-9001-1|title=Universes of Fuzzy Sets and Axiomatizations of Fuzzy Set Theory. Part II: Category Theoretic Approaches|journal=Studia Logica|volume=84|pages=23–50|year=2006|s2cid=10453751)) [https://web.archive.org/web/20120930203747/http://www.uni-leipzig.de/~logik/gottwald/SL-univers2b.pdf preprint]..
* Grattan-Guinness, I. (1975) Fuzzy membership mapped onto interval and many-valued quantities. Z. Math. Logik. Grundladen Math. 22, pp.&nbsp;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.&nbsp;325–332 
* Gylys, R. P. (1994) Quantal sets and sheaves over quantales, Liet. Matem. Rink., v. 34, No. 1, pp.&nbsp;9–31.
* ((cite book|editor=Ulrich Höhle, Stephen Ernest Rodabaugh|title=Mathematics of fuzzy sets: logic, topology, and measure theory|year=1999|publisher=Springer|isbn=978-0-7923-8388-8|series=The Handbooks of Fuzzy Sets Series|volume=3))
* ((cite journal |doi=10.1002/MANA.19750680109 |title=Intervall-wertige Mengen |date=1975 |last1=Jahn |first1=K.-U. |journal=Mathematische Nachrichten |volume=68 |pages=115–132 ))
* [[Kaufmann, Arnold]]. ''Introduction to the theory of fuzzy subsets.'' Vol. 2. Academic Pr, 1975.
* ((Cite book|last=Kerre|first=E.E.|chapter=A First View on the Alternatives of Fuzzy Set Theory |title=Computational Intelligence in Theory and Practice|editor1=B. Reusch|editor2=K-H. Temme|publisher=Physica-Verlag|location=Heidelberg|isbn=978-3-7908-1357-9|date=2001|pages=55–72|doi=10.1007/978-3-7908-1831-4_4))
* ((cite book|author1=George J. Klir|author2=Bo Yuan|title=Fuzzy sets and fuzzy logic: theory and applications|year=1995|publisher=Prentice Hall|isbn=978-0-13-101171-7))
* ((cite news|last=Kuzmin|first=V.B.|title=Building Group Decisions in Spaces of Strict and Fuzzy Binary Relations|location=Nauka, Moscow|date=1982|language=ru))
* Lake, J. (1976) [http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.130.6757&rep=rep1&type=pdf Sets, fuzzy sets, multisets and functions], J. London Math. Soc., II Ser., v. 12, pp.&nbsp;323–326 
* Meng, D., Zhang, X. and Qin, K. [https://www.sciencedirect.com/science/article/pii/S0898122111009138 Soft rough fuzzy sets and soft fuzzy rough sets], 'Computers & Mathematics with Applications', v. 62, issue 12, 2011, pp.&nbsp;4635–4645
* ((cite book |doi=10.1007/3-540-45523-X_11 |chapter=Fuzzy Multisets and Their Generalizations |title=Multiset Processing |series=Lecture Notes in Computer Science |date=2001 |last1=Miyamoto |first1=Sadaaki |volume=2235 |pages=225–235 |isbn=978-3-540-43063-6 ))
* Molodtsov, O. (1999) Soft set theory – first results, Computers & Mathematics with Applications, v. 37, No. 4/5, pp.&nbsp;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.&nbsp;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. [https://link.springer.com/chapter/10.1007/978-3-540-24844-6_78 On L-Fuzzy Rough Sets], Artificial Intelligence and Soft Computing – ICAISC 2004, 7th International Conference, Zakopane, Poland, June 7–11, 2004, Proceedings; 01/2004 
* ((cite journal|last=Salii|first=V.N.|date=1965|url=http://www.mathnet.ru/links/3c2c5808bf86871be61c7003d1efad97/ivm2487.pdf |title=Binary L-relations|journal=Izv. Vysh. Uchebn. Zaved. Matematika|volume=44|issue=1|pages=133–145|language=ru))
* 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) [https://www.tandfonline.com/doi/pdf/10.1007/s12543-011-0064-y 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) [https://www.researchgate.net/profile/Sabu_Sebastian/publication/220531149_Multi-Fuzzy_Extensions_of_Functions/links/56d320d308ae4d8d64a77552.pdf 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: [https://books.google.com/books?id=rdYRdlM2dAQC&dq=%22The+Genesis+of+Fuzzy+Set+Theory+and+Its+Initial+Applications%E2%80%94Developments+up+to+the+1970s%22&pg=PA1 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.&nbsp;165–235 
* Werro, Nicolas: [https://diuf.unifr.ch/main/is/sites/diuf.unifr.ch.main.is/files/documents/publications/WerroN.pdf Fuzzy Classification of Online Customers], University of Fribourg, Switzerland, 2008, [https://www.springer.com/cda/content/document/cda_downloaddocument/9783319159690-c2.pdf?SGWID=0-0-45-1495016-p177269846 Chapter 2] 
* Yager, R. R. (1986) On the Theory of Bags, International Journal of General Systems, v. 13, pp.&nbsp;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.&nbsp;301–321, 1997.
* Y. Y. Yao, A comparative study of fuzzy sets and rough sets, Information Sciences, v. 109, Issue 1–4, 1998, pp.&nbsp;227 – 242 
* Zadeh, L. (1975) [https://pdfs.semanticscholar.org/74d5/e9445c8154c8a77cc71487a4c5cfd9a9588c.pdf The concept of a linguistic variable and its application to approximate reasoning]–I, Inform. Sci., v. 8, pp.&nbsp;199–249
* ((cite book|author=Hans-Jürgen Zimmermann|title=Fuzzy set theory—and its applications|year=2001|publisher=Kluwer|isbn=978-0-7923-7435-0|edition=4th))

((refend))

((Non-classical logic))
((Set theory))

((DEFAULTSORT:Fuzzy Set))
[[Category:Fuzzy logic]]
[[Category:Systems of set theory]]

Pages transcluded onto the current version of this page (help):

Fuzzy set (edit)
Template:Ambox (view source) (template editor protected)
Template:Catalog lookup link (view source) (template editor protected)
Template:Category handler (view source) (protected)
Template:Citation needed (view source) (protected)
Template:Cite book (view source) (protected)
Template:Cite conference (view source) (protected)
Template:Cite journal (view source) (protected)
Template:Cite news (view source) (protected)
Template:Cite web (view source) (protected)
Template:Delink (view source) (protected)
Template:Div col (view source) (template editor protected)
Template:Div col/styles.css (view source) (template editor protected)
Template:Div col end (view source) (template editor protected)
Template:Doi (view source) (template editor protected)
Template:Fix (view source) (protected)
Template:Fix/category (view source) (protected)
Template:Harv (view source) (template editor protected)
Template:Harvard citation (view source) (template editor protected)
Template:Harvard citation no brackets (view source) (template editor protected)
Template:Harvard citation text (view source) (template editor protected)
Template:Harvnb (view source) (template editor protected)
Template:Harvtxt (view source) (template editor protected)
Template:Hlist (view source) (protected)
Template:Hlist/styles.css (view source) (protected)
Template:ISBN (view source) (template editor protected)
Template:Main (view source) (template editor protected)
Template:Main other (view source) (protected)
Template:Math (view source) (template editor protected)
Template:More citations needed (view source) (template editor protected)
Template:More citations needed section (view source) (template editor protected)
Template:Navbox (view source) (template editor protected)
Template:Non-classical logic (edit)
Template:Nowrap (view source) (protected)
Template:Pagetype (view source) (protected)
Template:Refbegin (view source) (template editor protected)
Template:Refbegin/styles.css (view source) (template editor protected)
Template:Refend (view source) (template editor protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:SDcat (view source) (protected)
Template:Set theory (edit)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Sp (view source) (template editor protected)
Template:Webarchive (view source) (template editor protected)
Template:Yesno (view source) (protected)
Template:Yesno-no (view source) (template editor protected)
Template:Yesno-yes (view source) (template editor protected)
Module:Arguments (view source) (protected)
Module:Catalog lookup link (view source) (template editor protected)
Module:Category handler (view source) (protected)
Module:Category handler/blacklist (view source) (protected)
Module:Category handler/config (view source) (protected)
Module:Category handler/data (view source) (protected)
Module:Category handler/shared (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Check isxn (view source) (template editor protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:Delink (view source) (protected)
Module:Disambiguation/templates (view source) (protected)
Module:Footnotes (view source) (template editor protected)
Module:Footnotes/anchor id list (view source) (template editor protected)
Module:Footnotes/anchor id list/data (view source) (template editor protected)
Module:Footnotes/whitelist (view source) (template editor protected)
Module:Format link (view source) (template editor protected)
Module:GetParameters (view source) (template editor protected)
Module:Hatnote (view source) (template editor protected)
Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:List (view source) (protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:SDcat (view source) (protected)
Module:String (view source) (protected)
Module:String2 (view source) (template editor protected)
Module:TableTools (view source) (protected)
Module:Unsubst (view source) (protected)
Module:Webarchive (view source) (template editor protected)
Module:Webarchive/data (view source) (template editor protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

Return to Fuzzy set.

Retrieved from "https://en.wikipedia.org/wiki/Fuzzy_set"