@misc{10481/90753,
year = {2024},
url = {https://hdl.handle.net/10481/90753},
abstract = {The fuzzy sets were introduced by Zadeh in [19]. Despite his proposal has been
facing harsh criticism in the previous years, this topic has recently grown exponentially,
mainly due to the possibilities of applicability of fields such decision
making or pattern recognition to applied science.
Although a great part of the publications related to this topic are focused on its
applications, the amount of work that connect fuzzy sets with the mathematics'
foundations are important too. For example, this is important for the Goguen's
catogories in [8] and for the classical set's theory. This proyect focuses on the
proposal to define the element of a fuzzy set that appears in [18].
In the first place, the basic notions in the fuzzy set's theory are defined, where
an order relation and the next operations: union (or supreme) and intersection
(or infimum) are established, all of them generalize the operations of the universe
set and the negation, which is a substitution of the subset's complement
concept. Likewise, the morphism definition between fuzzy sets appears. Then, it
is considerated a generalization of the concept when, the applications from the
unvierse set,, X to [0; 1] are replaced by any bonded lattice L, so the L-fuzzy
sets definition is obtained. Some examples of these extensions are widely used,
for instance the interval-valued fuzzy sets, where the lattice L is the lattice of all
closed interval contained in [0; 1].
In the second place this proyect describes the α-cuts os a fuzzy set, that is, a crisp
subsets' family of the universe set that verify some well-known properties which
are described in the Proposition 1.5.6. Throughtout these properties introduce
us into chapter 2 with the gradual set concept. This chapter starts characterizing
those subsets' family of X that determine a fuzzy set uniquelly, in other words,
those sets are a family of α-cuts exactly. One of these families can be interpreted
such as a map from the interval [0; 1] to the set P(X), which allows us to state the
gradual set definition. In this respect there are several proposal in the literature,
on the one hand the first by Dubois and Prade in [2] and on the other hand the
Wu's proposal in [18], which will be examined in this proyect.
The diference among the proposes is that Dubois and Prade set up as domain
the [0,1] interval in the first proposal, whle Wu admites any I ⊆ [0; 1] subset as
domain. The gradual element notion is defined from the gradual set notion by replacing
the P(X) codomain by the X set, i.e, a gradual element can be indentified with
a gradual set which image is a singleton (∀ α ∈ I).
Continuedly, it is focused on analysing the fuzzy element proposal that appears
in [18]. The study of how the fuzzy sets' operations can be recovered from the
fuzzy element notion constitutes the essential part of this proyect. It should be
noted that, the fuzzy element definition must be established throughout a gradual
element, that is, a map g : I ⊆ [0; 1] → X while a fuzzy set is described by a
map µ : X → [0; 1]. The most essential is to establish the enviroment where
both structures can be compared.
It will be understood that the fuzzy set which are determinated by a gradual
set is a fuzzy element os a certain fuzzy set (A; µA) if each α 2 IA g(α) 2 ∈α,
with IA the interval where the α-cuts of the fuzzy set A aren't empty. To de ne
a fuzzy element intervene the associating a gradual set to each fuzzy set process
and the reciprocal process.
The next issue covered is the proofs elabotation about how a gradual element
allows to recover the union (or supreme) and the intersectio (or infimus) description
of a finite fuzzy sets's family. There are references where some esed
results which deviate from the content os this proyect (Theorems 3.1.8 and
3.1.10) considerably can be consulted. In addition, the associativity property,
under corresponthing hypotheses, for the union and intersection defined is established.
Lastly, it's appear a problem: the negation definition. A fuzzy set's negation
through the fuzzy element concept is developed. In this case, it has been detected
that the definition which was proposed by Wu isn't consistent, and it's offered
an exam`ple about this cirsumstance.
It should be kept in mind that during all sections development, a special effort
has been made to add examples which help to understand and to clarify all the
concepts aand results.
The organization os this proyect is as follow.
Chapter 1 presents the basic fuzzy sets' theory definitions. It places special emphasis
on results which will be used in the following sections.
Second chapter establishes the relationship between fuzzy and gradual sets. These
will be the most important matter in the next chapter.
Finally, third chapter develops the proposal for defining the element of a fuzzy
set and its ability to determine operations such as the union and intersection of
a finite fuzzy sets' collection. The weaknesses about fuzzy element are analyzed All the objectives set out on the initial proyect's proposal've been developed,
even it has been carried out new challengues that had not been considered.},
organization = {Universidad de Granada. Facultad de Ciencias. Grado en Matemáticas. Trabajo Fin de Grado. Curso académico 2021-2022},
publisher = {Universidad de Granada},
title = {Sobre el concepto de elemento de un conjunto difuso},
author = {Cordovilla Sánchez, Celia},
}