Questions tagged [fuzzy-set]
For questions related to fuzzy set theory
82 questions
2
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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 ...
- 61
2
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0
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66
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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 ...
- 79
0
votes
1
answer
33
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If $\mu_a = \nu_a$ for all $a \in [0, 1]$, then $\mu = \nu$
Let $X$ be any nonempty set, let $\mu, \nu \colon X \longrightarrow [0, 1]$ be functions such that, for all $a \in [0, 1]$, we have
$$
\mu_a = \nu_a,
$$
where
$$
\mu_a := \{ x \in X \vert \mu(x) \geq ...
- 27.2k
0
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0
answers
15
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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 ...
- 1
0
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0
answers
19
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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 ...
- 321
1
vote
1
answer
197
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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 &...
- 627
1
vote
1
answer
252
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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 ...
- 627
1
vote
1
answer
71
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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$...
- 7,217
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1
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84
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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{...
- 7,217
0
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1
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49
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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 ...
- 109
1
vote
0
answers
69
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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, ...
- 11
1
vote
1
answer
437
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 ...
2
votes
0
answers
91
views
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 $A$...
- 21
0
votes
1
answer
570
views
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 ...
- 292
0
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0
answers
855
views
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 $\tilde A$...
- 445
1
vote
1
answer
261
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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 ...
- 399
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votes
0
answers
99
views
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 ...
- 498
0
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0
answers
88
views
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?
- 37
3
votes
1
answer
213
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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)=...
- 928
2
votes
0
answers
102
views
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 ...
- 12.1k
1
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0
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24
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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 ...
- 11
2
votes
0
answers
121
views
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 ...
- 121
2
votes
1
answer
251
views
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 ...
- 2,004
0
votes
1
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103
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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
- 35
0
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0
answers
48
views
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/...
- 33
1
vote
0
answers
28
views
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}
$$
if the inside of &...
2
votes
0
answers
64
views
What $f(A)$ means on this theorem?
I trying to understand this paper:
https://www.researchgate.net/publication/225302984_On_Intuitionistic_Anti-Fuzzy_Submodule_of_a_Module
and I want to prove this theorem:
If $f:M\to N$ be a ...
- 2,372
1
vote
1
answer
146
views
Existence and Uniqueness Proofs
I'm trying to prove the following theorem (and please, don't give me a proof, this is a conceptual question):
(Negoita and Ralescu's Representation Theorem) Let there be $A_{\alpha}$, $\alpha \in [0,1]...
- 464
0
votes
1
answer
204
views
Fuzzy sets property proof [closed]
I'm proving the distributive property for fuzzy sets and I'm having a bad time with the resolution of the following expression:
$$\max[\varphi_A(x),\min[\varphi_B(x),\varphi_C(x)]] = \\ \frac{1}{2}[\...
- 464
4
votes
1
answer
468
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 ...
- 105
2
votes
1
answer
182
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 ...
- 306
0
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0
answers
108
views
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
...
0
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0
answers
154
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 ...
2
votes
1
answer
551
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 \...
- 23
2
votes
0
answers
364
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. ...
- 303
1
vote
1
answer
343
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 ...
- 11
0
votes
1
answer
223
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. ...
- 151
5
votes
1
answer
215
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
$$
\{(x_0,x_1,\dots,x_n) \in \mathbb{R^{n+1}} |x_0+x_1+\dots+x_n = -\lg(a) \}
$...
0
votes
0
answers
73
views
Property of INF-w_i Composition of Fuzzy Relation
For any $\mathit a, b, d $ $\epsilon$ $\left[0,1\right]$, $\mathit a$ $\leq$ $\mathit b$ $\Rightarrow$ $$\\$$ i$\left.\right)$ $\omega_i$ $\left(a, d\right)$
$\geq$ $\omega_i$ $\left(b,d\right)$ $$\\...
0
votes
1
answer
50
views
How can i use fuzzy logic to switch between two distinct states?
So I have a nice problem and I've been contemplating the use of fuzzy logic for this. I have attached a little diagram I have made to explain the problem--> Fuzzy logic image
So I have a power demand ...
0
votes
0
answers
145
views
Applications of fuzzy logic/set theory in pure math?
I read in some posts on this website how fuzzy set theory is related to various reall life applications, e.g. computer programming, robotics, etc. I am wondering if anyone who is fairly into fuzzy set ...
- 3,830
1
vote
1
answer
982
views
A Question on Definition of Fuzzy Numbers
The fuzzy numbers are defined as fuzzy sets ($A$) defined over $\mathbb{R}$ which satisfy the following three properties:-
$A$ is normal, i.e., the height of $A$ is $1$.
$^{\alpha}A$ is a (non - ...
- 4,065
0
votes
1
answer
102
views
Proof from Fuzzy Intersection
While solving exercise of Fuzzy Sets and Fuzzy Logic: Theory and Applications by George J Klir and Bo Yuan, I came across this question:-
Let $i$ be a t - norm such that
$$i(a, b + c) = i(a, b) + i(...
- 4,065
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 ...
- 4,065
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 ...
- 133
1
vote
1
answer
149
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 ...
- 343
0
votes
1
answer
114
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 $$(A ...
- 89
1
vote
0
answers
443
views
What is the difference between probability theory and possibility theory?
This question I encountered when I was solving theoretical probability question. It was harder for me to find the exact difference between both. I guess some fuzzy logic term are also involved.
If ...
0
votes
1
answer
687
views
Proving Monotonicity of t-norm
For a lecture task I am trying to prove the monotonicity of a t-norm;
$$
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) \...
- 343
0
votes
1
answer
2k
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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 ...
- 187