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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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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 "()" is a matrix, does anyone know the meaning of …
A.A. Gde Jenana Putra
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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 fuzzy scale that transforms linguistic preferences…
clopez
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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 - empty) closed interval $\forall \alpha \in \left( 0,…
Aniruddha Deshmukh
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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-cut is so important for such operations and why it is…
Etemon
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How do I calculate m(.) of a set if I don't know what the Bel(.) of its subsets are?

Let $X=\{ a, b, c, d \}$. Given the belief measure $Bel(\{b\})= 0.1 , Bel(\{a,b\})= 0.2, Bel(\{b,c\}) = 0.3, Bel(\{b,d\})=1$ , determine the coresponding basic assignment. I'm trying to use the formula $m(A) = \Sigma_{B \in P(A)} (-1)^{|A-B|}.…
dismal-audience
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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 $s_{es}(a,b) \leq s_{es}(c,d)$. So, i want to prove prove that if…
Agung Izzul Haq
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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$\left.\right)$ $\omega_i$ $\left(a, d\right)$ $\geq$ $\omega_i$ $\left(b,d\right)$ $$\\$$ ii$\left.\right)$ $\omega_i$ $\left(d,a\right)$…
A.Chakraborty
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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 $$i(a, b + c) = i(a, b) + i(a, c)$$ for all $a, b, c \in [0, 1]$ and $b + c…
Aniruddha Deshmukh
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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 degrees of membership sum to one. I want to get a crisp…
TheVoiceInMyHead
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Set theory implications

I'm pretty new to the world of fuzzy set theory, and I am trying to understand implications. So, I am wondering if someone help tell me if the following is correct. I am trying to find the minimum of: $$ H \rightarrow \lnot G $$ and any advice or…
ofithch79
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Why isn't a degree of membership of an element being zero equivalent to an element not belonging to that set?

I was reading about fuzzy sets on Wikipedia. These sets are sets in which elements have a degree of membership ranging from $[0,1]$ and are defined by a set and a function mapping each element belonging to it to a degree of membership. This means…
IHaveAQuestion
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