Talk:Fuzzy set

Some topics

Citing from the article: A fuzzy set is a pair $(A,m)$ where $A$ is a set and $m:A\rightarrow [0,1]$. For each $x\in A$, $m(x)$ is called the grade of membership of $x$ in $(A,m)$.

If m is defined on A (i.e. takes as input all of the members of A) then it will never return 0. Accordingly, "Let $x\in A$. Then $x$ is called not included in the fuzzy set" does not make much sense, as does "The set $\{x\in A\mid m(x)>0\}$ is called the support of $(A,m)$" (seeing that $m(x)>0$ for every x in A).

A counter-argument would be to say that every element is contained in A. Seeing that this leads to there only being exactly one set, encompassing everything, this does not seem to be a good idea. If posing the question whether this set would contain itself, we arrive at the fuzzy equivalent to Russel's Paradox. Illuminations showing that I am mistaken are, of course, very welcome.

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What is the meaning of $A\wedge m(x)$ in the second line of the Definition section? $A$ is a set and $m(x)$ is a number. The minimum of these two doesn't make sense to me. Also later in this section what is the definition of $z_{1}$ etc? --Abel1981 18:19, 10 August 2007 (UTC)

The expression $A\wedge m(x)$ is not well-formed on it's own, because you cannot apply the AND operation to non-boolean values, however, the larger expression is well-formed: $x\in (A,m)\iff x\in A\wedge m(x)\neq 0$ which is equivalent to $x\in (A,m)\iff (x\in A)\wedge (m(x)\neq 0)$. --anon-Engineer —Preceding unsigned comment added by 128.118.40.77 (talk) 21:46, 27 September 2007 (UTC)

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Fuzzy sets are an extension of the classical set theory used in fuzzy logic.

They are intended to be, but they are not! Membership functions are just that - functions. Functions are defined in terms of sets. So the definition of a fuzzy set depends on (and therefore is not an extension of) the definition of a set. Blaise 21:53, 28 Apr 2005 (UTC)

According to Fuzzy set theory, classical sets are a form of fuzzy sets. This is not so otherway around. In a way the statment at the start of page is true. User:srinivasasha 18:07, 15 Apr 2005 (IST)

Blaise, (in the event that you ever read this) you may want to consider deleting your additions to several entries stating that fuzzy sets do not extend ordinary sets. First, axiomatizations of fuzzy sets which do not `depend on' ordinary set theory exist since 1967 (i.e. 2 years after Zadeh's definition and 38 years before you were here). Second, membership functions are a convenient model of fuzzy logic but they are not essential, as aptly shown in Petr Hajek's work. Third, even if one restricts oneself to the `popular' view of membership functions, your argument is a fallacious one. You say: Membership functions are functions, therefore special sets. My reply is: a set is something that satisfies the axioms of set theory, so it is something that cannot be considered isolated from the notions appearing in those axioms, like those of element, union, and so on. It is clear that fuzzy sets, *the way Fuzzy Set Theory regards them*, do not generally fulfil those axioms and so are generally not sets.

For example, in FST the union of two fuzzy sets is not the union of the graphs of their membership functions; the fact that unions of graphs of membership functions do fulfil the axioms of set theory is completely immaterial.

Another example, if A is a non-crisp fuzzy subset of a universe U, then, no matter how you define an `element', the axiom stating that two sets are equal when they have the same elements will fail (because then A would equal the ordinary set of its elements, which *is* crisp).155.210.232.88 16:47, 4 October 2006 (UTC)

>First, axiomatizations of fuzzy sets which do not `depend on' ordinary set theory exist since 1967

Good. Show me a reference.

>Second, membership functions are a convenient model of fuzzy logic but they are not essential,

This is the same as your first point.

>Third, even if one restricts oneself to the `popular' view of membership functions, your argument is a fallacious one. You say: Membership functions are functions, therefore special sets.

No, I say membership functions are functions, as generally understood in mathematics.

>My reply is: a set is something that satisfies the axioms of set theory, so it is something that cannot be considered isolated from the notions appearing in those axioms, like those of element, union, and so on. It is clear that fuzzy sets, *the way Fuzzy Set Theory regards them*, do not generally fulfil those axioms and so are generally not sets.

Agreed. What's that got to do with the point I was making, that fuzzy set theory isn't a generalisation of regular set theory because the notion of a membership function involves the notion of a function, which involves the notion of a set in the classical sense? Blaise 11:24, 15 September 2007 (UTC)

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(Possibly different IP address, but same person as 4 Oct 06.)

References.

The first full-fledged ZF-like axiomatization of fuzzy sets seems to be

E. William Chapin. Set-valued set theory. Part I: Notre Dame J. Formal Logic Volume 15, Number 4 (1974), 619-634. Part II: Notre Dame J. Formal Logic Volume 16, Number 2 (1975), 255-267 (free access at Project Euclid.)

1967 is the date of Goguen's JMAA paper cited in the main text, which is the first category-theoretical approach at fuzzy sets.

An overview of the many approaches is

S. Gottwald. Universes of fuzzy sets and axiomatizations of fuzzy set theory. Part I: Model-based and axiomatic approaches. Studia Logica 82, no.2 (2006), 211--244. Part II: Category theoretic approaches. Studia Logica 84 (2006), 23--50 (free access at Gottwald's website, and I suggest that they should be cited in the main text.)

Repetitions.

My first and second points are actually different. The first one concerns fuzzy sets, saying that axiomatic fuzzy set theories exist which do not rely on non-fuzzy set theories (as has been shown at your request.) The second one concerns fuzzy logic, saying that fuzzy logic makes full sense without any reference to fuzzy sets in the ordinary membership-function fashion (e.g. Hajek's basic logic in his book Metamathematics of fuzzy logic.) Admittedly, the second point is not central to the discussion, I guess I wanted to show that membership functions are not essential to either fuzzy set theory (point 1) or fuzzy logic (point 2), so there is no `backdoor' through which using fuzzy logic reintroduces the membership (set-theoretical) functions disposed of before.

Functions as sets.

I was taught that functions are subsets of the Cartesian product. I thought you meant that, but nothing changes if you're not in the `function=graph' boat.

What does it have to do?

My contention is not that your point is wrong, but that it is no point at all. Therefore, it is natural that I say things which do not revolve around it.--155.210.234.246 (talk) 04:10, 22 January 2008 (UTC)

Yes, Gottwald's two-part survey paper is good. I found it myself while browsing his list of publications. I'll add it to the further reading section, now that we have that. Tijfo098 (talk) 16:44, 13 April 2011 (UTC)

Fuzzy sets and Multi sets

Fuzzy sets are sets of pairs whereas a multiset is a pair $(X,m)$ where $X$ is a set and $m:X\rightarrow \mathbb {N}$. Should fuzzy sets be defined as multisets are, with an indicator function $m:X\rightarrow \mathbb {R}$ with a range $[0,1]$ or should multisets be defined as fuzzy sets are!? InformationSpace 03:05, 16 March 2007 (UTC).

I now have both Zadeh's original paper introducing fuzzy sets and a book Introduction to fuzzy sets, fuzzy logic, and fuzzy control systems. I can see from these that it is completely wrong to define a fuzzy set as a set of pairs. I guess I will have to correct this. InformationSpace 04:30, 17 July 2007 (UTC)

Ok, I've done this now. Although I like having a diagram, the one that was there had a number of problems - a fuzzy set (A,m) should be a subset of the crisp set A - some inconsistencies and omissions remain. No time right now... InformationSpace 05:05, 17 July 2007 (UTC)

Please explain! It makes no sense to declare that "a fuzzy set (A,m) should be a subset of the crisp set A" without providing a clear sense in which the notion "subset" extends to the union of {sets} and {fuzzy sets}. Without such, the nearest I can approach such a subset relation is this:

The set A of the fuzzy set (A,m) is a superset — not a subset — of the crisp set $A'=m^{-1}\{0,1\}$. yoyo (talk) 04:34, 5 October 2009 (UTC)

$\subseteq ,\cup$

I notice that there is no definition of $\subseteq$ or $\cup$ etc. w.r.t. a fuzzy sets. Someone should add these and other important fuzzy set operators to this article!InformationSpace 06:00, 4 May 2007 (UTC)

Fuzzy Mathematics

I just ran across this article on new pages patrole, but I'm way to unexperienced in this field to get any proper sources or better content for the article. It seems related to fuzzy sets though (the mathematics applied to fuzzy sets?). I posted an expert tag there, but maybe some of the editors of this page have a better idea what to do with it. (maybe a redirect? maybe a reverse redirect? keeping the article as a seperate one with seperate content?) Thanks. Martijn Hoekstra 21:54, 4 November 2007 (UTC)

An uncertain set

What is "an uncertain set"? yoyo (talk) 05:46, 5 October 2009 (UTC)

I think this is a typo and should refer to a fuzzy set. Can anyone confirm this? Also, what is meant by the set having "a mean interval"? acj (talk) 19:53, 3 September 2010 (UTC)

universal algebra - poset structure ?

The last paragraph in the Definition section is:

Sometimes, a more general definition is used, where membership functions take values in an arbitrary fixed algebra or structure L; usually it is required that L be at least a poset or lattice. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This generalization was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.

Algebra here refers to a universal algebra which has no inherited poset or even lattice structure. If anyone can post a link to the original paper I might be able to figure out the right more general definition. —Preceding unsigned comment added by The tree stump (talk • contribs) 15:40, 29 November 2009 (UTC)

Maybe my formulation in the article was misleading. Goguen in his 1967 paper already discusses several kinds of structures for membership degrees, ranging from posets to complete lattice-ordered semigroups. Later, various authors used various kinds of structures for membership degrees, from as general as any relation (e.g., here) to as specific as the real unit interval endowed with a particular set of operations (e.g., here). Most usually, the assumed structure of degrees is a lattice (often expanded by additional operations—e.g., a residuated lattice), or at least a poset (possibly satisfying some constraints). If it is a lattice, then the ordering relation can be represented just by the lattice operations of join and meet, in which case it is actually an algebra rather than a more general structure.

Any improvement of the formulation is of course welcome (I'll start with changing the singular "a more general definition" to plural, which may be less misleading). -- LBehounek (talk) 16:23, 7 December 2009 (UTC)

Klaua

The original paper of (Von) Dieter Klaua doesn't seem to be online. A later (1966) paper by him is though doi:10.1002/mana.19670330503. Klaua lived in East Germany. Tijfo098 (talk) 16:18, 13 April 2011 (UTC)

Siegfried Gottwald recently analyzed Klaua's 1965 paper in doi:10.1016/j.fss.2009.12.005. preprint Tijfo098 (talk) 16:22, 13 April 2011 (UTC)