Skip to main content
3 votes

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-...
gigalord's user avatar
  • 321
3 votes

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 ...
Ittay Weiss's user avatar
  • 80.1k
3 votes

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 ...
Nacho's user avatar
  • 167
2 votes

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 ...
Javier F. Botía's user avatar
2 votes

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 ...
Alex Ortiz's user avatar
  • 25.1k
2 votes
Accepted

Verifying De Morgan's laws for two given fuzzy sets

I guess that the univrse is $[0,\infty]$ because the membership function has to be between $0$ and $ 1$. Since $$f_A=\frac1{1+2x}\leq f_B=\frac1{\sqrt{1+2x}},\ \text{ if } x\geq 0$$ we have that $$...
zoli's user avatar
  • 20.5k
2 votes

A Question on Definition of Fuzzy Numbers

For condition 2 to be sufficient, it would need to imply condition 3. But surely this isn't true: suppose $A(x)=\exp(-x^2)$; then condition 2 holds but condition 3 doesn't.
Gareth McCaughan's user avatar
2 votes
Accepted

Existence and Uniqueness Proofs

One way to show existence is by construction. For instance, you might want to show that for any triple $(a, b, c)$ of integers with $$ 0 \le a < 2\\ 0 \le b < 3\\ 0 \le c < 5 $$ there's an ...
John Hughes's user avatar
2 votes
Accepted

On a definition of Spivak's fuzzy set

(Co)limits aren't just objects in a category: they're really (co)cones, which is an object equipped maps to or from other objects, which is universal with respect to this property. So $\operatorname{...
JHF's user avatar
  • 11k
1 vote
Accepted

upper semi-continuous of fuzzy set

Let $\mathbf{x} = \{x_i\}^n_{i=1}$. The indicator function of the unit ball defined as $$u(\mathbf{x}) = \begin{cases} 1 & \text{if $\Vert \mathbf{x} \Vert_2 = \left( \sum^{n}_{i=1} x_i^2 \right)^{...
user3733558's user avatar
  • 1,218
1 vote
Accepted

Equality of fuzzy sets

According to Jon Y. Kim , "Introduction to Fuzzy Set Theory" , Equality of Fuzzy Sets can be achieved easily , though the Initial Definition of Fuzzy Set itself requires a little tweaking ...
Prem's user avatar
  • 10.9k
1 vote

Why do we use $\alpha$-cut for arithmetic operations with fuzzy numbers?

The paper you referenced sorta starts the whole thing at the end. When we have real numbers, we can extend their operations to intervals. Each interval can be assigned a corresponding indicator ...
Gustavo de Souza's user avatar
1 vote
Accepted

In fuzzy sets, why do fraction notation, $\int$, and $+$ have different meanings than usual?

There are simply not enough symbols available to have a different symbol for every concept in all of mathematics. It is difficult to invent a new symbol or notation when writing a document ...
Ted's user avatar
  • 34.1k
1 vote

Randomness vs Fuzziness

Randomness is a concept of probability theory. It is not a concept of logic. Randomness means that there is a process that selects an element from a set. Each element of this set has an individual ...
Hubert Schölnast's user avatar
1 vote

Book request: fuzzy sets and logic

My favorite book on Fuzzy Sets and Fuzzy Logic is the text-book: George J. Klir and Bo Yuan: Fuzzy Sets and Fuzzy Logic: Theory and Applications (1995, Pearson Education) It is a well-written book and ...
Dr. Sundar's user avatar
  • 2,707
1 vote
Accepted

Fuzzy sets property proof

That’s doing it the hard way. Just prove that for any $x,y,z\in\Bbb R$, $$\max\big\{x,\min\{y,z\}\big\}=\min\big\{\max\{x,y\},\max\{x,z\}\big\}\,.\tag{1}$$ Without loss of generality we may assume ...
Brian M. Scott's user avatar
1 vote
Accepted

Why does supremum replace maximum in the generalisation?

Consider the example where we have $A_i(x)=-\frac 1 i$ for all $x$ in $X$ and for $i$ in $\mathcal I=\mathbb N$. Note that $\max[A_i(x),A_j(x)]=\max[-\frac 1i, -\frac 1j]$ is well defined. However $\...
Klaas van Aarsen's user avatar
1 vote
Accepted

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 ...
JMP's user avatar
  • 21.8k
1 vote
Accepted

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

$A \cap \overline{C} = \{(g,min(\mu_A(g), 1-\mu_C(g))) : g \in U \}$
William Elliot's user avatar
1 vote

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 ...
Charlie's user avatar
  • 151
1 vote
Accepted

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 ...
Quoka's user avatar
  • 3,131
1 vote
Accepted

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 $A_\alpha \cup B_\alpha$, ...
Zev Chonoles's user avatar
1 vote
Accepted

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 ...
Zhuoran He's user avatar
  • 3,039
1 vote
Accepted

Definition of linguistic variable.

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,...
Mauro ALLEGRANZA's user avatar
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 ...
Christian Blatter's user avatar
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 ...
Gspia's user avatar
  • 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 ...
Gspia's user avatar
  • 146

Only top scored, non community-wiki answers of a minimum length are eligible