Mixing
Fuzzy sets: extension principle
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I am a little bit confused about fuzzy set theory extension principle description. Most sources define extension principle in a following way:
Suppose $f:Xto Y$ and fuzzy set $Ain X$ such that $A=sum_{xin X}frac{mu_{A}(x)}{x}=frac{mu_{A}(x_{1})}{x_{1}}+...+frac{mu_{A}(x_{n})}{x_{n}}$.
Then extension principle states that the image of A under the mapping f can be expressed as a fuzy set $B=f(A)=frac{mu_{A}(f(x_{1}))}{x_{1}}+...+frac{mu_{A}(f(x_{n}))}{x_{n}}$
My question is how could fuzzy set be defined as a sum of rational numbers devided by vector? So in my consciousness it contradicts definition of fuzzy sets as a set of ordered pairs $(x,mu_{A}(x)), xin X$.
Will be very gratefull for explaining this problem or providing another, more easy to understand definition of extension principle!
fuzzy-logic fuzzy-set
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
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add a comment |
$begingroup$
I am a little bit confused about fuzzy set theory extension principle description. Most sources define extension principle in a following way:
Suppose $f:Xto Y$ and fuzzy set $Ain X$ such that $A=sum_{xin X}frac{mu_{A}(x)}{x}=frac{mu_{A}(x_{1})}{x_{1}}+...+frac{mu_{A}(x_{n})}{x_{n}}$.
Then extension principle states that the image of A under the mapping f can be expressed as a fuzy set $B=f(A)=frac{mu_{A}(f(x_{1}))}{x_{1}}+...+frac{mu_{A}(f(x_{n}))}{x_{n}}$
My question is how could fuzzy set be defined as a sum of rational numbers devided by vector? So in my consciousness it contradicts definition of fuzzy sets as a set of ordered pairs $(x,mu_{A}(x)), xin X$.
Will be very gratefull for explaining this problem or providing another, more easy to understand definition of extension principle!
fuzzy-logic fuzzy-set
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
$endgroup$
add a comment |
$begingroup$
I am a little bit confused about fuzzy set theory extension principle description. Most sources define extension principle in a following way:
Suppose $f:Xto Y$ and fuzzy set $Ain X$ such that $A=sum_{xin X}frac{mu_{A}(x)}{x}=frac{mu_{A}(x_{1})}{x_{1}}+...+frac{mu_{A}(x_{n})}{x_{n}}$.
Then extension principle states that the image of A under the mapping f can be expressed as a fuzy set $B=f(A)=frac{mu_{A}(f(x_{1}))}{x_{1}}+...+frac{mu_{A}(f(x_{n}))}{x_{n}}$
My question is how could fuzzy set be defined as a sum of rational numbers devided by vector? So in my consciousness it contradicts definition of fuzzy sets as a set of ordered pairs $(x,mu_{A}(x)), xin X$.
Will be very gratefull for explaining this problem or providing another, more easy to understand definition of extension principle!
fuzzy-logic fuzzy-set
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
$endgroup$
I am a little bit confused about fuzzy set theory extension principle description. Most sources define extension principle in a following way:
Suppose $f:Xto Y$ and fuzzy set $Ain X$ such that $A=sum_{xin X}frac{mu_{A}(x)}{x}=frac{mu_{A}(x_{1})}{x_{1}}+...+frac{mu_{A}(x_{n})}{x_{n}}$.
Then extension principle states that the image of A under the mapping f can be expressed as a fuzy set $B=f(A)=frac{mu_{A}(f(x_{1}))}{x_{1}}+...+frac{mu_{A}(f(x_{n}))}{x_{n}}$
My question is how could fuzzy set be defined as a sum of rational numbers devided by vector? So in my consciousness it contradicts definition of fuzzy sets as a set of ordered pairs $(x,mu_{A}(x)), xin X$.
Will be very gratefull for explaining this problem or providing another, more easy to understand definition of extension principle!
fuzzy-logic fuzzy-set
fuzzy-logic fuzzy-set
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
edited Apr 10 '16 at 13:50
Andrés E. Caicedo
65.5k8159250
65.5k8159250
asked Apr 10 '16 at 10:31
BogdanBogdan
709
asked Apr 10 '16 at 10:31
BogdanBogdan
709
asked Apr 10 '16 at 10:31
BogdanBogdan
709
709
add a comment |
add a comment |
1 Answer
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Klir & Folger [1] says that
$B=f(A)=frac{μ(x_1)}{f(x_1)}+...+frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.
answered Aug 4 '16 at 7:25
GspiaGspia
1114
$endgroup$
add a comment |
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1 Answer
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1 Answer
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active
oldest
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$begingroup$
Klir & Folger [1] says that
$B=f(A)=frac{μ(x_1)}{f(x_1)}+...+frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.
answered Aug 4 '16 at 7:25
GspiaGspia
1114
$endgroup$
add a comment |
$begingroup$
Klir & Folger [1] says that
$B=f(A)=frac{μ(x_1)}{f(x_1)}+...+frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.
answered Aug 4 '16 at 7:25
GspiaGspia
1114
$endgroup$
add a comment |
$begingroup$
Klir & Folger [1] says that
$B=f(A)=frac{μ(x_1)}{f(x_1)}+...+frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.
answered Aug 4 '16 at 7:25
GspiaGspia
1114
$endgroup$
Klir & Folger [1] says that
$B=f(A)=frac{μ(x_1)}{f(x_1)}+...+frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy subset $A$ to a fuzzy subset $B$.
If more than one element of $X$ is mapped by $f$ to the same element $y in Y$, then the maximum of the grades is chosen as grade for $y$ in $f(A)$.
[1] Klir & Folger, Fuzzy Sets, Uncertainty and Information, 1988.
answered Aug 4 '16 at 7:25
GspiaGspia
1114
edited Aug 13 '16 at 12:44
answered Aug 4 '16 at 7:25
GspiaGspia
1114
answered Aug 4 '16 at 7:25
GspiaGspia
1114
answered Aug 4 '16 at 7:25
GspiaGspia
1114
1114
add a comment |
add a comment |
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