Fuzzy sets are sets whose elements have degrees of membership. Fuzzy sets were introduced by Lotfi A. Zadeh (1965) as an extension of the classical notion of set.[1] In classical set theory, the membership of elements in a set is assessed in binary terms according to a bivalent condition — an element either belongs or does not belong to the set. By contrast, fuzzy set theory permits the gradual assessment of the membership of elements in a set; this is described with the aid of a membership function valued in the real unit interval [0, 1]. Fuzzy sets generalize classical sets, since the indicator functions of classical sets are special cases of the membership functions of fuzzy sets, if the latter only take values 0 or 1.[2] Classical bivalent sets are in fuzzy set theory 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 [3].
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A fuzzy set is a pair (A,m) where A is a set and ![m : A \rightarrow [0,1]](http://images-mediawiki-sites.thefullwiki.org/07/3/9/7/1398771271881225.png)
.
For each 
,
m(x) is called the
grade of membership of x in (A,m). For a finite set
A =
{x1,...,xn},
the fuzzy set (A,m)
is often denoted by {m(x1) /
x1,...,m(xn)
/ xn}.
Let .
Then x is called not
included in the fuzzy set (A,m) if m(x) = 0, x is called fully
included if m(x) =
1, and x is called
fuzzy member if 0 <
m(x) < 1.[4] The set
is called the support of (A,m) and the set 
is called its kernel.
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. The usual membership functions with values in [0, 1] are then called [0, 1]-valued membership functions. This kind of generalizations was first considered in 1967 by Joseph Goguen, who was a student of Zadeh.[5]
As an extension of the case of multi-valued logic, valuations (
)
of propositional variables (Vo) into a set
of membership degrees (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.[6]
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."[7]
Industrial applications of fuzzy sets in the context of "fuzzy logic in the wider sense" can be found at fuzzy logic.
A fuzzy number is a convex, normalized fuzzy set 
whose membership function is at least segmentally continuous and has the functional
value μA(x) =
1 at precisely one element. This can be likened to the funfair game "guess your
weight," where someone guesses the contestant's weight, with closer
guesses being more correct, and where the guesser "wins" if he or
she guesses near enough to the contestant's weight, with the actual
weight being completely correct (mapping to 1 by the membership
function).
A fuzzy interval is an uncertain set
with a mean interval whose elements possess the membership function
value μA(x) =
1. As in fuzzy numbers, the membership function must be convex, normalized, at least segmentally
continuous.[8]
   | Fuzzy Image Processing - Fuzzy Image Processing: Fuzzy Sets |
| | Uncertainty model Fuzziness - Fuzziness — Uncertainty in Engineering |
| | The Algorithm of Fuzzy Analysis - Fuzzy Analysis — Uncertainty in Engineering |
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