Fuzzy set

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 $[0,1]$ ${\ displaystyle [0,1]}$ $[0, 1]$ , not just the values $0$ ${\ displaystyle 0}$ ${\ displaystyle 0}$ or $one$ ${\ displaystyle 1}$ $one$ . It is the basic concept of fuzzy logic .

Definition

Under the fuzzy set $A$ ${\ displaystyle A}$ $A$ the set of ordered pairs made up of elements is understood $x$ ${\ displaystyle x}$ $x$ universal set $X$ ${\ displaystyle X}$ $X$ and corresponding degrees of affiliation $\mu _{A}(x)$ ${\ displaystyle \ mu _ {A} (x)}$ $\mu _{A}(x)$ :

A=\{(x,\mu _{A}(x))\mid x\in X\}

{\ displaystyle A = \ {(x, \ mu _ {A} (x)) \ mid x \ in X \}}

A=\{(x,\mu _{A}(x))\mid x\in X\}

,

moreover $\mu _{A}(x)$ ${\ displaystyle \ mu _ {A} (x)}$ $\mu _{A}(x)$ - membership function (generalization of the concept of a characteristic function of ordinary clear sets), indicating the degree to which an element $x$ ${\ displaystyle x}$ $x$ belongs to a fuzzy set $A$ ${\ displaystyle A}$ $A$ . Function $\mu _{A}(x)\$ ${\ displaystyle \ mu _ {A} (x) \}$ $\mu _{A}(x)\$ takes values in some linearly ordered set $M$ ${\ displaystyle M}$ $M$ . Lots of $M$ ${\ displaystyle M}$ $M$ called many accessories , often as $M$ ${\ displaystyle M}$ $M$ segment is selected $[0,1]$ ${\ displaystyle [0,1]}$ $[0, 1]$ . If a $M=\{0,1\}\$ ${\ displaystyle M = \ {0,1 \} \}$ $M=\{0,1\}\$ (that is, consists of only two elements), then the fuzzy set can be considered as an ordinary clear set.

Basic Definitions

Let be $A$ ${\ displaystyle A}$ $A$ fuzzy set with elements from the universal set $X\$ ${\ displaystyle X \}$ $X\$ and many accessories $M=[0,1]$ ${\ displaystyle M = [0,1]}$ $M=[0,1]$ . Then:

carrier ( support ) of a fuzzy set $\operatorname {supp} A$ ${\ displaystyle \ operatorname {supp} A}$ $\operatorname {supp} A$ called set $\{x\mid x\in X,\mu _{A}(x)>0\}$ ${\ displaystyle \ {x \ mid x \ in X, \ mu _ {A} (x)> 0 \}}$ $\{x\mid x\in X,\mu _{A}(x)>0\}$ ;
value $\sup _{x\in X}\mu _{A}(x)$ ${\ displaystyle \ sup _ {x \ in X} \ mu _ {A} (x)}$ $\sup _{{x\in X}}\mu _{A}(x)$ called the height of a fuzzy set $A\$ ${\ displaystyle A \}$ $A\$ . Fuzzy set $A\$ ${\ displaystyle A \}$ $A\$ ok if its height is equal $1\$ ${\ displaystyle 1 \}$ $1\$ . If the height is strictly less $1\$ ${\ displaystyle 1 \}$ $1\$ , a fuzzy set is called subnormal ;
a fuzzy set is empty if $\forall x\in X:\mu _{A}(x)=0$ ${\ displaystyle \ forall x \ in X: \ mu _ {A} (x) = 0}$ $\forall x\in X:\mu _{A}(x)=0$ . A non-empty subnormal fuzzy set can be normalized by the formula
$\mu '_{A}(x)={\frac {\mu _{A}(x)}{\sup \mu _{A}(x)}}$ ${\ displaystyle \ mu '_ {A} (x) = {\ frac {\ mu _ {A} (x)} {\ sup \ mu _ {A} (x)}}}$ $\mu '_{A}(x)={\frac {\mu _{A}(x)}{\sup \mu _{A}(x)}}$ ;
a fuzzy set is unimodal if $\mu _{A}(x)=1\$ ${\ displaystyle \ mu _ {A} (x) = 1 \}$ $\mu _{A}(x)=1\$ only on one $x\$ ${\ displaystyle x \}$ $x\$ of $X\$ ${\ displaystyle X \}$ $X\$ ;
the elements $x\in X$ ${\ displaystyle x \ in X}$ $x\in X$ for which $\mu _{A}(x)=0{,}5$ ${\ displaystyle \ mu _ {A} (x) = 0 {,} 5}$ $\mu _{A}(x)=0{,}5$ are called transition points of a fuzzy set $A\$ ${\ displaystyle A \}$ $A\$ .

Comparison of fuzzy sets

Let be $A$ ${\ displaystyle A}$ $A$ and $B$ ${\ displaystyle B}$ $B$ fuzzy sets defined on a universal set $X$ ${\ displaystyle X}$ $X$ .

$A$ ${\ displaystyle A}$ contained in $B$ ${\ displaystyle B}$ if for any element of $X$ ${\ displaystyle X}$ its membership function $A$ ${\ displaystyle A}$ will take a value less than or equal to the membership function $B$ ${\ displaystyle B}$ :
$A\subset B\Leftrightarrow \forall x\in X:\mu _{A}(x)\leqslant \mu _{B}(x)$ ${\ displaystyle A \ subset B \ Leftrightarrow \ forall x \ in X: \ mu _ {A} (x) \ leqslant \ mu _ {B} (x)}$ .
In case the condition $\mu _{A}(x)\leqslant \mu _{B}(x)$ ${\ displaystyle \ mu _ {A} (x) \ leqslant \ mu _ {B} (x)}$ not for everyone $x\in X$ ${\ displaystyle x \ in X}$ , talk about the degree of inclusion of the fuzzy set $A$ ${\ displaystyle A}$ at $B$ ${\ displaystyle B}$ , which is defined as follows:
$l\left(A\subset B\right)=\min _{x\in T}\mu _{B}(x)$ ${\ displaystyle l \ left (A \ subset B \ right) = \ min _ {x \ in T} \ mu _ {B} (x)}$ where $T=\{x\in X;\mu _{A}(x)\leqslant \mu _{B}(x),\mu _{A}(x)>0\}$ ${\ displaystyle T = \ {x \ in X; \ mu _ {A} (x) \ leqslant \ mu _ {B} (x), \ mu _ {A} (x)> 0 \}}$ .
Two sets are called equal if they are contained in each other:
$A=B\Leftrightarrow \forall x\in X:\mu _{A}(x)=\mu _{B}(x)$ ${\ displaystyle A = B \ Leftrightarrow \ forall x \ in X: \ mu _ {A} (x) = \ mu _ {B} (x)}$ .
In case the values of membership functions $\mu _{A}(x)$ ${\ displaystyle \ mu _ {A} (x)}$ and $\mu _{B}(x)$ ${\ displaystyle \ mu _ {B} (x)}$ almost equal to each other, they say about the degree of equality of fuzzy sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ , for example, in the form
$E(A=B)=1-\max _{x\in T}|\mu _{A}(x)-\mu _{B}(x)|$ ${\ displaystyle E (A = B) = 1- \ max _ {x \ in T} | \ mu _ {A} (x) - \ mu _ {B} (x) |}$ where $T=\{x\in X;\mu _{A}(x)\neq \mu _{B}(x)\}$ ${\ displaystyle T = \ {x \ in X; \ mu _ {A} (x) \ neq \ mu _ {B} (x) \}}$ .

Properties of fuzzy sets

$\alpha$ ${\ displaystyle \ alpha}$ -section of a fuzzy set $A\subseteq X$ ${\ displaystyle A \ subseteq X}$ denoted by $A_{\alpha }$ ${\ displaystyle A _ {\ alpha}}$ , the following distinct set is called:

A_{\alpha }=\{x\in X\mid \mu _{A}(x)\geqslant \alpha \}

{\ displaystyle A _ {\ alpha} = \ {x \ in X \ mid \ mu _ {A} (x) \ geqslant \ alpha \}}

,

that is, a set defined by the following characteristic function (membership function):

\chi _{A_{\alpha }}(x)=\left\{{\begin{matrix}0,&\mu _{A}(x)<\alpha ,\\1,&\mu _{A}(x)\geqslant \alpha .\end{matrix}}\right.

{\ displaystyle \ chi _ {A _ {\ alpha}} (x) = \ left \ {{\ begin {matrix} 0, & \ mu _ {A} (x) <\ alpha, \\ 1, & \ mu _ {A} (x) \ geqslant \ alpha. \ End {matrix}} \ right.}

For $\alpha$ ${\ displaystyle \ alpha}$ -section of a fuzzy set is true implication:

\alpha _{1}<\alpha _{2}\Rightarrow A_{\alpha _{1}}\supset A_{\alpha _{2}}

{\ displaystyle \ alpha _ {1} <\ alpha _ {2} \ Rightarrow A _ {\ alpha _ {1}} \ supset A _ {\ alpha _ {2}}}

.

Fuzzy set $A\subseteq \mathbf {R}$ ${\ displaystyle A \ subseteq \ mathbf {R}}$ is convex if and only if the condition is satisfied:

\mu _{A}[\gamma x_{1}+(1-\gamma )x_{2}]\geqslant \langle \mu _{A}(x_{1})\land \mu _{A}(x_{2})=\min\{\mu _{A}(x_{1}),\mu _{A}(x_{2})\}\rangle

{\ displaystyle \ mu _ {A} [\ gamma x_ {1} + (1- \ gamma) x_ {2}] \ geqslant \ langle \ mu _ {A} (x_ {1}) \ land \ mu _ { A} (x_ {2}) = \ min \ {\ mu _ {A} (x_ {1}), \ mu _ {A} (x_ {2}) \} \ rangle}

for any $x_{1},x_{2}\in \mathbf {R}$ ${\ displaystyle x_ {1}, x_ {2} \ in \ mathbf {R}}$ and $\gamma \in [0,1]$ ${\ displaystyle \ gamma \ in [0,1]}$ .

Fuzzy set $A\subseteq \mathbf {R}$ ${\ displaystyle A \ subseteq \ mathbf {R}}$ is concave if and only if the condition is satisfied:

\mu _{A}[\gamma x_{1}+(1-\gamma )x_{2}]\leqslant \langle \mu _{A}(x_{1})\lor \mu _{A}(x_{2})=\max\{\mu _{A}(x_{1}),\mu _{A}(x_{2})\}\rangle

{\ displaystyle \ mu _ {A} [\ gamma x_ {1} + (1- \ gamma) x_ {2}] \ leqslant \ langle \ mu _ {A} (x_ {1}) \ lor \ mu _ { A} (x_ {2}) = \ max \ {\ mu _ {A} (x_ {1}), \ mu _ {A} (x_ {2}) \} \ rangle}

for any $x_{1},x_{2}\in \mathbf {R}$ ${\ displaystyle x_ {1}, x_ {2} \ in \ mathbf {R}}$ and $\gamma \in [0,1]$ ${\ displaystyle \ gamma \ in [0,1]}$ .

Fuzzy Set Operations

With many accessories $M=[0,1]\$ ${\ displaystyle M = [0,1] \}$

Intersection of fuzzy sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ called a fuzzy subset with a membership function, which is a minimum of membership functions $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ :
$\mu _{A\cap B}(x)=\min(\mu _{A}(x),\mu _{B}(x))$ ${\ displaystyle \ mu _ {A \ cap B} (x) = \ min (\ mu _ {A} (x), \ mu _ {B} (x))}$ .
The product of fuzzy sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ called a fuzzy subset with membership function:
$\mu _{AB}(x)=\mu _{A}(x)\mu _{B}(x)$ ${\ displaystyle \ mu _ {AB} (x) = \ mu _ {A} (x) \ mu _ {B} (x)}$ .
The union of fuzzy sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ called a fuzzy subset with membership function, which is a maxim of membership functions $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ :
$\mu _{A\cup B}(x)=\max(\mu _{A}(x),\mu _{B}(x))$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ max (\ mu _ {A} (x), \ mu _ {B} (x))}$ .
Sum of fuzzy sets $A$ ${\ displaystyle A}$ and $B$ ${\ displaystyle B}$ called a fuzzy subset with membership function:
$\mu _{A+B}(x)=\mu _{A}(x)+\mu _{B}(x)\ -\mu _{A}(x)\mu _{B}(x)$ ${\ displaystyle \ mu _ {A + B} (x) = \ mu _ {A} (x) + \ mu _ {B} (x) \ - \ mu _ {A} (x) \ mu _ {B } (x)}$ .
Negation of the set $A\$ ${\ displaystyle A \}$ called set ${\overline {A}}$ ${\ displaystyle {\ overline {A}}}$ with membership function:
$\mu _{\overline {A}}(x)=1-\mu _{A}(x)$ ${\ displaystyle \ mu _ {\ overline {A}} (x) = 1- \ mu _ {A} (x)}$ for each $x\in X$ ${\ displaystyle x \ in X}$ .

Alternative Representation of Operations on Fuzzy Sets

Intersection

In general, the operation of intersecting fuzzy sets is defined as follows:

\mu _{A\cap B}(x)=T(\mu _{A}(x),\mu _{B}(x))

{\ displaystyle \ mu _ {A \ cap B} (x) = T (\ mu _ {A} (x), \ mu _ {B} (x))}

,

where is the function $T$ ${\ displaystyle T}$ Is the so-called T-norm . The following are specific examples of the implementation of the T-norm :

$\mu _{A\cap B}(x)=\mu _{A}(x)\land \mu _{B}(x)=\min(\mu _{A}(x),\mu _{B}(x))$ ${\ displaystyle \ mu _ {A \ cap B} (x) = \ mu _ {A} (x) \ land \ mu _ {B} (x) = \ min (\ mu _ {A} (x), \ mu _ {B} (x))}$
$\mu _{A\cap B}(x)=\mu _{A}(x)\mu _{B}(x)$ ${\ displaystyle \ mu _ {A \ cap B} (x) = \ mu _ {A} (x) \ mu _ {B} (x)}$
$\mu _{A\cap B}(x)=\max\{0,\mu _{A}(x)+\mu _{B}(x)-1\}$ ${\ displaystyle \ mu _ {A \ cap B} (x) = \ max \ {0, \ mu _ {A} (x) + \ mu _ {B} (x) -1 \}}$
$\mu _{A\cap B}(x)=\left\{{\begin{matrix}\mu _{A}(x),&\mu _{B}(x)=1\\\mu _{B}(x),&\mu _{A}(x)=1\\0,&\mu _{A}(x)<1,\mu _{B}(x)<1,\end{matrix}}\right.$ ${\ displaystyle \ mu _ {A \ cap B} (x) = \ left \ {{\ begin {matrix} \ mu _ {A} (x), & \ mu _ {B} (x) = 1 \\ \ mu _ {B} (x), & \ mu _ {A} (x) = 1 \\ 0, & \ mu _ {A} (x) <1, \ mu _ {B} (x) <1 , \ end {matrix}} \ right.}$
$\mu _{A\cap B}(x)=1-\min\{1,[(1-\mu _{A}(x))^{p}+(1-\mu _{B}(x))^{p}]^{1 \over p}\}$ ${\ displaystyle \ mu _ {A \ cap B} (x) = 1- \ min \ {1, [(1- \ mu _ {A} (x)) ^ {p} + (1- \ mu _ { B} (x)) ^ {p}] ^ {1 \ over p} \}}$ for $p\geqslant 1$ ${\ displaystyle p \ geqslant 1}$

Association

In the general case, the operation of combining fuzzy sets is defined as follows:

\mu _{A\cup B}(x)=S(\mu _{A}(x),\mu _{B}(x))

{\ displaystyle \ mu _ {A \ cup B} (x) = S (\ mu _ {A} (x), \ mu _ {B} (x))}

,

where is the function $S$ ${\ displaystyle S}$ - T-conorm . The following are specific examples of the implementation of the S-norm :

$\mu _{A\cup B}(x)=\mu _{A}(x)\lor \mu _{B}(x)=\max(\mu _{A}(x),\mu _{B}(x))$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ mu _ {A} (x) \ lor \ mu _ {B} (x) = \ max (\ mu _ {A} (x), \ mu _ {B} (x))}$
$\mu _{A\cup B}(x)=\mu _{A}(x)+\mu _{B}(x)-\mu _{A}(x)\mu _{B}(x)$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ mu _ {A} (x) + \ mu _ {B} (x) - \ mu _ {A} (x) \ mu _ {B } (x)}$
$\mu _{A\cup B}(x)=\min\{1,\mu _{A}(x)+\mu _{B}(x)\}$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ min \ {1, \ mu _ {A} (x) + \ mu _ {B} (x) \}}$
$\mu _{A\cup B}(x)=\left\{{\begin{matrix}\mu _{A}(x),&\mu _{B}(x)=0\\\mu _{B}(x),&\mu _{A}(x)=0\\1,&\mu _{A}(x)>0,\mu _{B}(x)>0\end{matrix}}\right.$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ left \ {{\ begin {matrix} \ mu _ {A} (x), & \ mu _ {B} (x) = 0 \\ \ mu _ {B} (x), & \ mu _ {A} (x) = 0 \\ 1, & \ mu _ {A} (x)> 0, \ mu _ {B} (x)> 0 \ end {matrix}} \ right.}$
$\mu _{A\cup B}(x)=\min\{1,[\mu _{A}^{p}(x)+\mu _{B}^{p}(x)]^{1 \over p}\}$ ${\ displaystyle \ mu _ {A \ cup B} (x) = \ min \ {1, [\ mu _ {A} ^ {p} (x) + \ mu _ {B} ^ {p} (x) ] ^ {1 \ over p} \}}$ for $p\geqslant 1$ ${\ displaystyle p \ geqslant 1}$

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 $\mu _{A}(x)\$ ${\ displaystyle \ mu _ {A} (x) \}$ can be considered as the probability of covering an element $x\$ ${\ displaystyle x \}$ some random set $B\$ ${\ displaystyle B \}$ .

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 $X=\{x_{1},x_{2},x_{3},x_{4}\}$ ${\ displaystyle X = \ {x_ {1}, x_ {2}, x_ {3}, x_ {4} \}}$
many accessories $M=[0,1]$ ${\ displaystyle M = [0,1]}$
$A$ {\ displaystyle A} and $B$ {\ displaystyle B} - two fuzzy subsets $X$ {\ displaystyle X}
- $A=\{(x_{1}\mid 0{,}4),(x_{2}\mid 0{,}6),(x_{3}\mid 0),(x_{4}\mid 1)\}$ ${\ displaystyle A = \ {(x_ {1} \ mid 0 {,} 4), (x_ {2} \ mid 0 {,} 6), (x_ {3} \ mid 0), (x_ {4} \ mid 1) \}}$
- $B=\{(x_{1}\mid 0{,}3),(x_{2}\mid 0),(x_{3}\mid 0),(x_{4}\mid 0{,}2)\}$ ${\ displaystyle B = \ {(x_ {1} \ mid 0 {,} 3), (x_ {2} \ mid 0), (x_ {3} \ mid 0), (x_ {4} \ mid 0 { ,} 2) \}}$

Results of the main operations:

intersection: ${A\cap B}=\{(x_{1}\mid 0{,}3),(x_{2}\mid 0),(x_{3}\mid 0),(x_{4}\mid 0{,}2)\}={B}$ ${\ displaystyle {A \ cap B} = \ {(x_ {1} \ mid 0 {,} 3), (x_ {2} \ mid 0), (x_ {3} \ mid 0), (x_ {4 } \ mid 0 {,} 2) \} = {B}}$
Union: ${A\cup B}=\{(x_{1}\mid 0{,}4),(x_{2}\mid 0{,}6),(x_{3}\mid 0),(x_{4}\mid 1)\}={A}$ ${\ displaystyle {A \ cup B} = \ {(x_ {1} \ mid 0 {,} 4), (x_ {2} \ mid 0 {,} 6), (x_ {3} \ mid 0), (x_ {4} \ mid 1) \} = {A}}$

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.

[1] Bulletin of the Academy of Sciences of the Georgian SSR . - Academy, 1974. - S. 157. - 786 p.

[2] AM Shirokov. Fundamentals of the theory of acquisition . - Science and Technology, 1987. - S. 66. - 198 p.

[3] 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 .

[4] Chemistry and life, XXI century . - The company "Chemistry and Life", 2008. - S. 37. - 472 p.