# Questions tagged [fuzzy-set]

For questions related to fuzzy set theory

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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 ...
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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 ...
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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 &...
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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 ...
1 vote
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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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### 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 ...
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1 vote
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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, ...
1 vote
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### 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 ...
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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 $A$...
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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 ...
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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 $\tilde A$...
1 vote
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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 ...
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### 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. ...
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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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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 - ...
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### 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 ...
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### 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) \...
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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 ...
### 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 \...