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Questions tagged [fuzzy-set]

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Quantifying the distance between two discrete fuzzy sets

I am looking to use fuzzy sets to represent several collections of data points. Then, given a crisp set, I'd like to determine which collection the crisp set is most similar to. Each collection is ...
Alex 's user avatar
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Proving the minimum t-norm is a joint possibility distribution

A t-norm is an operator $T:[0,1]^2\rightarrow[0,1]$ which is comutative, monotonic, associative and has 1 as an identity element, that is, $T(1,x)=T(x,1)=x$. A joint possibility distribution (JPD) of ...
Gustavo de Souza's user avatar
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Interpretation of "If A, then B" as "A coupled with B": Context and Applicability

It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that &...
hasanghaforian's user avatar
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Equality of fuzzy sets

Is there a definition for "equality of fuzzy sets" ? My current thinking : Say we have two fuzzy sets $A = \{(x,\mu_{A}(x)):x \in X\}$ and $B = \{(y,\mu_{B}(y)):y \in Y\}$ When we consider ...
hasanghaforian's user avatar
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Why do we use $\alpha$-cut for arithmetic operations with fuzzy numbers?

I just started studying fuzzy sets. In the context of fuzzy numbers, I saw the arithmetic operations are defined with respect to $\alpha$-cut (For example see this paper). But I don't know why $\alpha$...
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In fuzzy sets, why do fraction notation, $\int$, and $+$ have different meanings than usual?

I have just started to learn about fuzzy sets from this website which is written in Persian. Here are some quotations, If $U$ is a finite set, we usually denote the fuzzy set $A$ as $$ A=\left\{\frac{...
Etemon's user avatar
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Is there a way to find anomalies in fuzzy time series?

To clarify, A fuzzy number $p$ is called L-R fuzzy number if its membership function $\mu_{p}$ can be expressed through reference functions $L$ and $R$ in the form $$ \mu_{p} (x) = \begin{...
hjbello's user avatar
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Proving a property about alpha cut $ (A')_\alpha \neq (A_\alpha)', unless \\ \alpha = 0.5 $

I recently came accross this property about alpha (or lambda) cuts $$ (A')_\alpha \neq (A_\alpha)', unless \\ \alpha = 0.5 $$ where A is a fuzzy set with membership function $\mu_A(x)$ I am curious ...
Jay's user avatar
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The definition of fuzzy sets using logical and set operators.

I wrote this definition of fuzzy sets and fuzzy logic for a college assignment and was wondering if it is correct: Let a be any ordinary element of the universal superset U and A be a subset of U, ...
Matt Schramm's user avatar
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341 views

First decomposition theorem of fuzzy set

I read if standard union in fuzzy set have definition: Union of two fuzzy sets $\tilde{A}$ and $\tilde{B}$ in universe $X$ denoted $\tilde{A}\cup\tilde{B}$ is fuzzy set in universe $X$ with membership ...
Leudofikus De Ferento's user avatar
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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 $A$...
Amine's user avatar
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Randomness vs Fuzziness

As the title suggests, What is the difference between randomness and fuzziness? My take: They are two-sides of the same coin - they are two different ways of expressing uncertainty. Consider a ...
x.projekt's user avatar
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Meaning of cardinality of fuzzy sets & intuitionistic fuzzy sets

We know that the cardinality of a finite crisp (or, classical) set $A$ can be considered as a measure of "number of elements" of $A$. However, if $X$ is a universe of discourse and $\tilde A$...
Usual_Learner's user avatar
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1 answer
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Book request: fuzzy sets and logic

There are requests for this topic already, though I am looking for a particular kind of book on the topic. I got part of the way through Trillas' and Eciolaza's book Fuzzy Logic: An Introductory ...
Mark's user avatar
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Monotonicity of Einstein Sum (s-norm)

I am trying to prove that Einstein Sum $$S_{es}(a,b) = \frac{a+b}{1+ab}$$ is an s-norm operator. But, i got struggle on proving its monotonicity, i.e If $a\leq c$ and $b \leq d$ then $s_{es}(a,b) \leq ...
Agung Izzul Haq's user avatar
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Which branch of mathematics is the fuzzy logic?

Fuzzy logic comes close to boolean algebra but is the upper branch of fuzzy logic (or fuzzy mathematics) still algebra?
Max Hager's user avatar
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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)=...
masaheb's user avatar
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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 ...
Greg Nisbet's user avatar
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Fuzzy Logic Composition

I want to know when to use max-min composition and max product composition. I'm pretty sure I understand how to compute them, but I notice that even though they're both supposed to be performing ...
J J's user avatar
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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 ...
carlo's user avatar
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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 ...
Math.mx's user avatar
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upper semi-continuous of fuzzy set [closed]

Let $u:\mathbb{R^n}\to [0,1]$ be a fuzzy set. (fuzzy set is a set of ordered pairs $(x,u(x)), x\in \mathbb{R^n})$. Please give an example such that $u(x)$ be upper semi-continuous. thanks
user809500's user avatar
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Non continuous Fuzzy Set

I stuck in this question for a few hours now. Can anyone help me? $ A ̃ $ is a non continuous fuzzy set and is defined by: $ A ̃ = \{0.6/x_1 +0.5/x_2 + 1/x_3 + 0.75/x_4 + 0.7/x_5 + 0.8/x_6 + 1/...
XiaLihu's user avatar
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The meaning of this equation

I've read a paper Obstacle Avoidance Approaches for Autonomous Underwater Vehicle and I found the equation like : $$ \alpha = \max_i \min_k \left(R \triangleleft R^T \right)_{ik} $$ if the inside of &...
A.A. Gde Jenana Putra's user avatar
2 votes
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64 views

What $f(A)$ means on this theorem?

I trying to understand this paper: https://www.researchgate.net/publication/225302984_On_Intuitionistic_Anti-Fuzzy_Submodule_of_a_Module and I want to prove this theorem: If $f:M\to N$ be a ...
Ongky Denny Wijaya's user avatar
1 vote
1 answer
142 views

Existence and Uniqueness Proofs

I'm trying to prove the following theorem (and please, don't give me a proof, this is a conceptual question): (Negoita and Ralescu's Representation Theorem) Let there be $A_{\alpha}$, $\alpha \in [0,1]...
Fractal Admirer's user avatar
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1 answer
191 views

Fuzzy sets property proof [closed]

I'm proving the distributive property for fuzzy sets and I'm having a bad time with the resolution of the following expression: $$\max[\varphi_A(x),\min[\varphi_B(x),\varphi_C(x)]] = \\ \frac{1}{2}[\...
Fractal Admirer's user avatar
4 votes
1 answer
437 views

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 ...
Jayitha Reddy's user avatar
2 votes
1 answer
177 views

Division by $0$ Extreme Case in Fuzzy C-Means Clustering

I have a question about calculating the partition matrix for the Fuzzy C-Means (FCM) Clustering Algorithm. For any point $x_i$ and cluster centroid $c_j$, the membership value $w_{i,j}$ is computed by ...
rhkoulen's user avatar
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Composition of a fuzzy set

I'm trying to learn about Fuzzy alone. I'm having difficulties with understanding intuitively the composition of a fuzzy set. According to the definition in the book:enter image description here ...
Rəşad Abdulxalıqov's user avatar
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149 views

Uncertain Set theory: example and axiom clarification

I am working through the work of Baoding Liu on uncertainty theory, specifically uncertain sets. The concept was introduced to represent 'unsharp' concepts such as humans being 'tall' or 'young' or a ...
Jordan MacLachlan's user avatar
2 votes
1 answer
527 views

How can I compute the complement of a mathematical membership function?

Considering fuzzy set A defined on real numbers by the membership functions: $\mu_A(x)=\frac{1}{x+1}, \mu_C(x)=\frac{1}{10^x}$ How can I determine mathematical membership function and graph of $ A \...
estamos's user avatar
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2 votes
0 answers
326 views

Intuition behind Zadeh / Lukasiewicz implication of fuzzy logic

I am studying fuzzy logic and the implication operator. Suppose we are given the implication "IF $x$ is $A$ THEN $y$ is B" with $A,B$ fuzzy sets so $A(x), B(x) \in [0,1]$ are the membership functions. ...
NightRain23's user avatar
1 vote
0 answers
293 views

Calculating a fuzzy crisp value from a linguistic fuzzy weight

I am struggling to find a clear source of information on-line that will help me understand how to convert a fuzzy weight for a linguistic preference to a crisp value. For instance, below we have a ...
clopez's user avatar
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0 votes
1 answer
189 views

Example of membership function which does not equal 1 for any element in its domain

I'm learning about fuzzy logic and fuzzy sets, and it seems to me that there is no requirement that there be at least one element in the domain set for which the membership function is equal to 1. ...
Charlie's user avatar
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5 votes
1 answer
202 views

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) \} $...
ComplecialSimplex's user avatar
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59 views

Property of INF-w_i Composition of Fuzzy Relation

For any $\mathit a, b, d $ $\epsilon$ $\left[0,1\right]$, $\mathit a$ $\leq$ $\mathit b$ $\Rightarrow$ $$\\$$ i$\left.\right)$ $\omega_i$ $\left(a, d\right)$ $\geq$ $\omega_i$ $\left(b,d\right)$ $$\\...
A.Chakraborty's user avatar
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1 answer
43 views

How can i use fuzzy logic to switch between two distinct states?

So I have a nice problem and I've been contemplating the use of fuzzy logic for this. I have attached a little diagram I have made to explain the problem--> Fuzzy logic image So I have a power demand ...
Akash Menon's user avatar
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0 answers
138 views

Applications of fuzzy logic/set theory in pure math?

I read in some posts on this website how fuzzy set theory is related to various reall life applications, e.g. computer programming, robotics, etc. I am wondering if anyone who is fairly into fuzzy set ...
Squirrel-Power's user avatar
1 vote
1 answer
912 views

A Question on Definition of Fuzzy Numbers

The fuzzy numbers are defined as fuzzy sets ($A$) defined over $\mathbb{R}$ which satisfy the following three properties:- $A$ is normal, i.e., the height of $A$ is $1$. $^{\alpha}A$ is a (non - ...
Aniruddha Deshmukh's user avatar
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1 answer
100 views

Proof from Fuzzy Intersection

While solving exercise of Fuzzy Sets and Fuzzy Logic: Theory and Applications by George J Klir and Bo Yuan, I came across this question:- Let $i$ be a t - norm such that $$i(a, b + c) = i(a, b) + i(...
Aniruddha Deshmukh's user avatar
1 vote
0 answers
2k views

Operations on Fuzzy Power sets

With reference to a question:Operations on power set, I wanted to generalize it for a power set of fuzzy sets. So, first let us try to define power set of fuzzy sets. Let $\mathscr{F}(X)$ be the set ...
Aniruddha Deshmukh's user avatar
2 votes
1 answer
5k views

Verifying De Morgan's laws for two given fuzzy sets [closed]

How does one show graphically or otherwise that two fuzzy sets A and B with membership functions $1/(1+2x)$ and $1/\sqrt{1+2x}$ respectively, satisfy De Morgan's laws? This is a question from Neural ...
Sasha R's user avatar
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1 vote
1 answer
146 views

Comma Notation Meaning in Fuzzy Logic Statement

I have the following statement; $$ (A \cup B)_\alpha = A_\alpha \cup B_\alpha, (A \cap B)_\alpha = A_\alpha \cap B_\alpha $$ The minimum and maximum operators represent the intersection and union of ...
James Izzard's user avatar
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1 answer
111 views

Proving that $(A \cap ((B \cap C) \cup (A^c \cap C^c)) \cup C^c= (A \cap B \cap C) \cup C^c$

I currently started learning about fuzzy sets as well as operators and functions that can be applied to them. As such, I came by this question: If $A, B, C$ are an element of $F$, then show that $$(A ...
a22asin's user avatar
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1 vote
0 answers
430 views

What is the difference between probability theory and possibility theory?

This question I encountered when I was solving theoretical probability question. It was harder for me to find the exact difference between both. I guess some fuzzy logic term are also involved. If ...
Kim Jong Un's user avatar
0 votes
1 answer
666 views

Proving Monotonicity of t-norm

For a lecture task I am trying to prove the monotonicity of a t-norm; $$ T_H(x,y)=\frac{x\cdot y}{x +y -xy} $$ So I interpret this as being required to demonstrate that; $$ T(x,y) \leq T(x,z) \...
James Izzard's user avatar
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1 answer
2k views

Help with alpha-cuts in fuzzy sets

basically all I need to know is what are the standard methods to achieve the below. So, I have a fuzzy set A containing (say) four elements. For each element I have a degree of membership. The ...
TheVoiceInMyHead's user avatar
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1 answer
183 views

Find the inverse of a function $f : X \rightarrow Y$ between two fuzzy topological spaces $X$ and $Y$?

Suppose $(X, \tau_1), (Y, \tau_2)$ be two fuzzy topological spaces, where, $X=\{a\}, Y=\{x, y\}, \tau_1 =\{0_X, 1_X, \{(a, 0.3)\}\}, \tau_2 =\{0_Y, 1_Y, \{(x, 0.2), (y, 0.2)\}\}, $ and $f:X \...
Birojit Das's user avatar
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0 answers
64 views

How are the bar and colon used in this function?

A search on fuzzy logic led me to a Zadeh function in the context of "synthesis of fuzzy logic functions given in tabular form". The following expression is defined as a row of a choice table of a ...
DukeZhou's user avatar
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