Fuzzy set (sometimes blurry [1] , vague [2] , foggy [3] , furry [4] ) is a concept introduced by Lotfi Zadeh in 1965 in the article “Fuzzy Sets” in the journal , in which expanded the classical concept of a set , assuming that the characteristic function of the set (called the Zade function of membership for a fuzzy set) can take any values in the interval , not just the values or . It is the basic concept of fuzzy logic .
Definition
Under the fuzzy set the set of ordered pairs made up of elements is understood universal set and corresponding degrees of affiliation :
- ,
moreover - membership function (generalization of the concept of a characteristic function of ordinary clear sets), indicating the degree to which an element belongs to a fuzzy set . Function takes values in some linearly ordered set . Lots of called many accessories , often as segment is selected . If a (that is, consists of only two elements), then the fuzzy set can be considered as an ordinary clear set.
Basic Definitions
Let be fuzzy set with elements from the universal set and many accessories . Then:
- carrier ( support ) of a fuzzy set called set ;
- value called the height of a fuzzy set . Fuzzy set ok if its height is equal . If the height is strictly less , a fuzzy set is called subnormal ;
- a fuzzy set is empty if . A non-empty subnormal fuzzy set can be normalized by the formula
- ;
- a fuzzy set is unimodal if only on one of ;
- the elements for which are called transition points of a fuzzy set .
Comparison of fuzzy sets
Let be and fuzzy sets defined on a universal set .
- contained in if for any element of its membership function will take a value less than or equal to the membership function :
- .
- In case the condition not for everyone , talk about the degree of inclusion of the fuzzy set at , which is defined as follows:
- where .
- Two sets are called equal if they are contained in each other:
- .
- In case the values of membership functions and almost equal to each other, they say about the degree of equality of fuzzy sets and , for example, in the form
- where .
Properties of fuzzy sets
-section of a fuzzy set denoted by , the following distinct set is called:
- ,
that is, a set defined by the following characteristic function (membership function):
For -section of a fuzzy set is true implication:
- .
Fuzzy set is convex if and only if the condition is satisfied:
for any and .
Fuzzy set is concave if and only if the condition is satisfied:
for any and .
Fuzzy Set Operations
With many accessories
- Intersection of fuzzy sets and called a fuzzy subset with a membership function, which is a minimum of membership functions and :
- .
- The product of fuzzy sets and called a fuzzy subset with membership function:
- .
- The union of fuzzy sets and called a fuzzy subset with membership function, which is a maxim of membership functions and :
- .
- Sum of fuzzy sets and called a fuzzy subset with membership function:
- .
- Negation of the set called set with membership function:
- for each .
Alternative Representation of Operations on Fuzzy Sets
Intersection
In general, the operation of intersecting fuzzy sets is defined as follows:
- ,
where is the function Is the so-called T-norm . The following are specific examples of the implementation of the T-norm :
- for
Association
In the general case, the operation of combining fuzzy sets is defined as follows:
- ,
where is the function - T-conorm . The following are specific examples of the implementation of the S-norm :
- for
Relation to Probability Theory
The theory of fuzzy sets in a certain sense reduces to the theory of random sets and thereby to probability theory . The basic idea is that the value of the membership function can be considered as the probability of covering an element some random set .
However, in practical application, the apparatus of the theory of fuzzy sets is usually used independently, competing with the apparatus of probability theory and applied statistics . For example, in control theory there is a direction in which fuzzy sets (fuzzy controllers) are used instead of probability theory methods to synthesize expert controllers .
Examples
Let be:
- lots of
- many accessories
- and - two fuzzy subsets
Results of the main operations:
- intersection:
- Union:
Notes
- ↑ Bulletin of the Academy of Sciences of the Georgian SSR . - Academy, 1974. - S. 157. - 786 p.
- ↑ AM Shirokov. Fundamentals of the theory of acquisition . - Science and Technology, 1987. - S. 66. - 198 p.
- ↑ Kozlova Natalia Nikolaevna. Color picture of the world in the language // Scientific notes of the Transbaikal State University. Series: Philology, History, Oriental Studies. - 2010. - Issue. 3 . - ISSN 2308-8753 .
- ↑ Chemistry and life, XXI century . - The company "Chemistry and Life", 2008. - S. 37. - 472 p.
Literature
- Zade L. The concept of a linguistic variable and its application to making approximate decisions. - M .: Mir, 1976 .-- 166 p.
- Kofman A. Introduction to the theory of fuzzy sets. - M .: Radio and communications, 1982.- 432 p.
- Fuzzy sets and theory of possibilities: Recent achievements / R. R. Jager. - M .: Radio and communications, 1986.
- Zadeh LA Fuzzy sets // Information and Control. - 1965. - T. 8 , No. 3 . - P. 338-353.
- Orlovsky S. A. Decision-making problems with fuzzy source information. - M .: Nauka, 1981. - 208 p. - 7600 copies.