Questions tagged [fuzzy-set]
For questions related to fuzzy set theory
81 questions
1 vote
0 answers
52 views
About Fuzzy Logic [closed]
I am reading fuzzy logic and completed basics like fuzzy sets , fuzzy arithmetic , operations and other things. I want to study advanced topics like Interval type 2 Fuzzy sets, ordered fuzzy numbers ...
4 votes
1 answer
133 views
Why is $(g∘f)^{-1} (A)=A∘(g∘f)$?
Thm: Let $(x,τ)$ be a fuzzy topological space. The identity function $f:(X,τ)→(X,τ)$ is fuzzy continuous. Proof: Let $A∈τ$. Since $f^{-1}(A)=A(f)=A∘f=A$, then $f^{-1} (A)∈τ$. The same method is used ...
- 1,148
2 votes
0 answers
53 views
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 ...
- 71
2 votes
0 answers
80 views
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
68 views
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 $$ \mu_a := \{ x \in X \vert \mu(x) \geq ...
- 27.8k
1 vote
1 answer
273 views
Interpretation of "If A, then B" as "A coupled with B": Context and Applicability
It is known that the meaning of a conditional statement in fuzzy logic can vary depending on the interpretation and context. In certain fuzzy logic books, I have come across the interpretation that &...
- 783
1 vote
1 answer
333 views
Equality of fuzzy sets
Is there a definition for "equality of fuzzy sets" ? My current thinking : Say we have two fuzzy sets $A = \{(x,\mu_{A}(x)):x \in X\}$ and $B = \{(y,\mu_{B}(y)):y \in Y\}$ When we consider ...
- 783
1 vote
1 answer
138 views
Why do we use $\alpha$-cut for arithmetic operations with fuzzy numbers?
I just started studying fuzzy sets. In the context of fuzzy numbers, I saw the arithmetic operations are defined with respect to $\alpha$-cut (For example see this paper). But I don't know why $\alpha$...
- 8,489
0 votes
1 answer
110 views
In fuzzy sets, why do fraction notation, $\int$, and $+$ have different meanings than usual?
I have just started to learn about fuzzy sets from this website which is written in Persian. Here are some quotations, If $U$ is a finite set, we usually denote the fuzzy set $A$ as $$ A=\left\{\frac{...
- 8,489
0 votes
1 answer
100 views
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 ...
- 113
1 vote
0 answers
83 views
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
540 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
118 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
692 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 ...
- 302
0 votes
0 answers
956 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 $\tilde A$...
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