# Tag Info

### Why does Spivak define the realization functor from fuzzy simplicial sets to extended pseudo metric spaces the way he does?

One possible explanation of "why $-lg$?" is that he needed a way to transform fuzzy set strengths, which range from $(0, 1]$, to metric space distances, which range from $[0, \infty)$, in a non-...
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### Is Fuzzy Set/Measure Theory an Active Area for Research?

The appeal you speak of, which I believe is/was shared by many, is that the fuzzy variants of the traditional mathematical objects are, intuitively, more realistic. After all, in real life we never ...
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### Need help to understand LAMDA clustering algorithm

It would be useful to know your reference. This algorithm is not popular and there is a lack of details on its original creation (J Aguilar-Martin and R De Mantaras. 1982. The process of ...
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### Need help to understand LAMDA clustering algorithm

I've worked the LAMDA algorithm since 2009 and I recommend reading the following paper where my brother and me propose a novel LAMDA algorithm: Javier Fernando Botía Valderrama, Diego José Luis Botía ...

### Is there such a thing as a weighed relation?

There is such a thing as a fuzzy set that is used in the context of machine learning. A fuzzy set $(X, \chi)$ has a characteristic function $\chi: X \to [0,1]$, and thus does not admit strictly binary ...
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Accepted

• 5,908
1 vote
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### Division by $0$ Extreme Case in Fuzzy C-Means Clustering

This is a special case of the theorem where it is assumed that no $c_k=x_i$. The original paper this formula appeared in is: A Fuzzy Relative of the ISODATA Process and Its Use in Detecting Compact ...
• 21.8k
1 vote
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### How can I compute the complement of a mathematical membership function?

$A \cap \overline{C} = \{(g,min(\mu_A(g), 1-\mu_C(g))) : g \in U \}$
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### Example of membership function which does not equal 1 for any element in its domain

Actually the answer is yes... For example, consider the set of tennis ball colours in a 4-ball canister. Are they fluoro yellow or fluoro green...? The colour of each ball - arguably - has non-zero ...
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### Proof from Fuzzy Intersection

We will prove the statement for $i(a,b)$ where $b$ is rational Since the rationals are dense in $[0,1]$, you can use monotonicity of $t$-norms to generalize this to all $b\in [0,1]$. To see what we ...
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### Comma Notation Meaning in Fuzzy Logic Statement

It is a comma as used in everyday English, used to separate clauses / statements. Nothing specific to math. It rained today, and I bought a hat. $(A \cup B)_\alpha$ equals $A_\alpha \cup B_\alpha$, ...
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1 vote
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### Proving Monotonicity of t-norm

Rewrite the t-norm as $$T_H(x,y)=\frac{1}{x^{-1}+y^{-1}-1}.$$ So it increases monotonically with both $x$ and $y\in(0,1]$. When dealing with t-norms, try to put them into a form in which the ...
• 3,039
1 vote
Accepted

See Linguistic variables and truth-values in fuzzy logic : A linguistic variable $\mathcal X$ is a nonfuzzy variable which ranges over a collection $T (\mathcal X)$ of structured fuzzy variables $X_1,... • 94.6k 1 vote ### How should I refer to "volume-like" measures in a dimensional-free way? If the "intrinsic" dimension of an object$O$is$d\geq0$then its$d'$-dimensional Hausdorff measure is$=\infty$when$d'<d$, and is$=0$when$d'>d$. If you are lucky its$d$-dimensional ... 1 vote Accepted ### Fuzzy sets: extension principle Klir & Folger [1] says that$B=f(A)=\frac{μ(x_1)}{f(x_1)}+...+\frac{μ(x_n)}{f(x_n)}$, that is, the extension principle applies maps that map a crisp set to another, to fuzzy sets, turning a fuzzy ... • 146 1 vote ### Given a fuzzy set, find the level set Klir & Folger [1] say that$\alpha$-cut of a fuzzy set$A$is a crisp set$A_\alpha$that contains the elements that have a membership grade in$A$greater than or equal to$\alpha\$. And the set ...
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