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Questions tagged [fuzzy-set]

For questions related to fuzzy set theory

In fuzzy set theory, elements have varying degrees of membership in sets.

Formally, a fuzzy set is a pair $(S,m)$ where $S$ is a set and $m\colon S\to[0.1]$. $m$ is a membership function. If $m(x)=0$, the element $x$ is not included in the set. If $m(x)=1$, then $x$ is fully included in the set.

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Differential forms on fuzzy manifolds

This post will take a bit to set up properly, but it is an easy read (and most likely easy to answer); in any event, please bear with me. Question In the usual setting of open subsets of $\mathbb{R}^n$, differential forms are defined as…
Isaac Sledge
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Is Fuzzy Set/Measure Theory an Active Area for Research?

I came across the notion of a fuzzy set the other day and since then, I've been reading about fuzzy measures and the Sugeno/Choquet integrals. While I certainly do not claim to have fully wrapped my mind around it by any means, there is something…
AnalysisStudent
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Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?

In Spivak's paper on metric realization of fuzzy simplicial sets, he sends a fuzzy $n$-simplex of strength $a$ to the set $$ \{(x_0,x_1,\dots,x_n) \in \mathbb{R^{n+1}} |x_0+x_1+\dots+x_n = -\lg(a) \} $$ and the realization of a general $X$ is via…
ComplecialSimplex
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A symmetric function

While working on a research problem on fuzzy metric spaces, I came across a special symmetric function $F_n:X^n\times (0,\infty)\to [0,1]$ i.e. \begin{equation*} F_n(x_1,x_2,\dots,x_n,t)=F_n(x_{\pi(1)},x_{\pi(2)},...,x_{\pi(n)},t)…
K A Khan
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Why is $(g∘f)^{-1} (A)=A∘(g∘f)$?

Thm: Let $(x,τ)$ be a fuzzy topological space. The identity function $f:(X,τ)→(X,τ)$ is fuzzy continuous. Proof: Let $A∈τ$. Since $f^{-1}(A)=A(f)=A∘f=A$, then $f^{-1} (A)∈τ$. The same method is used in Thm: A composition of fuzzy continuous…
Isaac
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Why does supremum replace maximum in the generalisation?

I've recently taken on interest for Fuzzy Set Theory and I've been reading George J. Klir and Bo Yuan. 1994. Fuzzy sets and fuzzy logic: theory and applications. Prentice-Hall, Inc., USA. Where the authors define the standard union of two fuzzy sets…
Jayitha Reddy
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Prove that two fuzzy sets are disjoint if and only if their supports are disjoint

Prove that two fuzzy sets are disjoint if and only if their supports are disjoint. Given two fuzzy sets $A,B$ of a reference set $X$,then : $$ \begin{align} \\ &\text{Supp}(A) \cap \text{Supp}(B)= \emptyset\\ &\iff \nexists x \in X:x \in…
masaheb
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Understanding Fuzzy Composition operations

There are two common forms of composition operation in Fuzzy Theory: max–min composition max–product composition Let R be a relation that relates elements from universe X to universe Y, and let S be a relation that relates elements from universe…
Atinesh Singh
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Troubles figuring out how to calculate the centroid for a defuzzification process

I am trying to implement a fuzzy logic system, but am having serious issues finding the centroid for the defuzzification process. This is what my output sets look like: My reference source gives this to me as an example: This is going to sound…
theJuls
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Asking Help for Continuity on Fuzzy Topological Space and its Proof Verification

I am reading the paper Fuzzy Topological Spaces and Fuzzy Compactness by Robert Lowen. Lowen didn't wrote down his proof about proposition 3.1 since he thought it is trivial. But I would like to ask some help for the verification for my proof. Given…
Teh Ais Kaw
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Asking help for a proof verification in the paper "Fuzzy Topological Spaces and Fuzzy Compactness" by R-Lowen

I am reading the paper Fuzzy Topological Spaces and Fuzzy Compactness by Robert Lowen. I have proved the theorem 2.2: $(X,\delta)$ is topologically generated if and only if for each continuous function $f \in \mathscr{G}(I,I_r)$ and for each $\nu…
I like Milo
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How to find Zadeh's extension of a function like this?

I'm learning fuzzy logic and i don't find many examples explaining Zadeh's extension principle I found this one but i don't know how to solve it. Can you help me ? Let us consider two fuzzy subsets $A$ and $B$ defined by their membership functions…
Amine
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Is almost-naive set theory in fuzzy logic with comprehension limited to continuous connectives consistent?

I've heard the result before that naive set theory is consistent in infinite-valued Łukasiewicz logic. This answer contains a citation. In this logic, every connective is continuous (w.r.t the product topology when necessary). Additionally, $[0, 1]$…
Greg Nisbet
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help replicating fuzzy equations from a paper

I'm trying to replicate Zhou's Paper on quantifying UX using Fuzzy Math. In their model, there is a weight vector $A$ for a set of characteristics. in the paper's test case the characteristics were Effectiveness, Efficiency, and Satisfaction with…
carlo
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On a definition of Spivak's fuzzy set

In the paper "Metric Realization of Fuzzy Simplicial Sets" of David Spivak it takes $I=(0,1]$ as poset and consider it as a category. He gives it a Grothendieck topology induce it from consider $I$ as topological space with topology induced from…
Math.mx
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