Lianget al.in 2000 defined interval type-2 fuzzy sets (IT2FSs), which constitute a subset of type-2 fuzzy sets. While the membership degrees in the former are functions from [0, 1] to [0, 1] (fuzzy truth values), the membership degrees in IT2FSs only take their values in$\lbrace {\text{0}},{\text{1}}\rbrace$. Although all the initial work on IT2FSs involved convex membership degrees only, in 2015, Bustinceet al.began the study on IT2FSs in general, including certain sets with nonconvex membership degrees. However, these are obviously early stages, with a lot of open problems regarding the theoretical structure of IT2FSs. For example, as far as we know, no negation operator has been obtained in this context. Therefore, it seems appropriate to continue with the study started in previous papers, delving deeper into the properties and operations of IT2FSs. Consequently, this work studies the structure of the set of functions from [0, 1] to$\lbrace {\text{0}},{\text{1}}\rbrace$(expanding the set considered by Bustinceet al.), from which we have removed the constant function$\mathbf{0}$, to offer a different study to the one carried out by Walker and Walker. More specifically, we consider join and meet operations, partial order derived from each one, and the negation operators in that set. Among other results, we provide new characterizations of join and meet operations and of partial orders on the set of functions from [0, 1] to$ \lbrace {\text{0}},{\text{1}} \rbrace $; we also present the first negation operators on this set.

Type-2 fuzzy sets (T2FSs) were introduced by L. A. Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). In this extension, the degree to which an element belongs to a set is just a label of the linguistic variable “TRUTH,” which allows to represent reality in a more appropriate way. On the other hand, negations play an essential role within fuzzy sets theory. In fact, they are necessary in order to obtain, for example, complements of fuzzy sets, dual of at-norm or a t-conorm, entropies, implications, as well as to study the possible contradictions appearing in a fuzzy system. However, meanwhile negations on [0, 1] (set of the membership degrees of a fuzzy set) have been deeply studied throughout the literature, the same has not happened with the negations on M = Map ([0, 1], [0, 1]), set of functions from [0, 1] to [0, 1] (and also set of the membership degrees of a T2FS), and so many aspects of the negations on M have not yet been investigated. In a previous paper, the axioms that an operation must satisfy to be considered a negation or a strong negation on a bounded partially ordered set were established. A family of strong negations on L and set of normal and convex functions of M were also presented. Moreover, let us note that the main characteristic of fuzzy systems is just the flexibility in order to be able to represent knowledge according to each situation, offering different models among which the expert can choose the one that best suits his/her criteria. Thus, it seems useful to find broad sets of negations in M and in L, which, as far as we know, have not been done by other researchers. According to these ideas, in this paper, the authors first present new negations and strong negations on L, and then show, for the first time, some negations on M with respect to each of the two partial orders defined in this set.

Type-2 fuzzy sets (T2FSs) were introduced by Zadeh in 1975 as an extension of type-1 fuzzy sets. The degree of membership of an element for T2FSs is a fuzzy set in [0, 1], that is, a T2FS is determined by a membership function from the universe of discourse X to M, where M is the set of functions from [0, 1] to [0, 1]. Walker and Walker extended the definitions oft-norm (triangular norm) and t-conorm to L (subset of normal and convex functions of M), establishing the tr-norms and tr-conorms (according to the “restrictive axioms” given by them), and defined two families of binary operations on M and found that, under certain conditions, these operations are tr.-norms or tr.-conorms on L. In this paper, we introduce more general binary operations on M than those given by Walker and Walker and study which of the minimum conditions necessary for these operations satisfy each of the axioms of the tr-norm and tr-conorm. In particular, interesting results about the closure properties are obtained, and the main result of the paper provides sufficient conditions for the given operations to be tr.-norms or tr-conorms on L.

This paper describes the use of the granular linguistic model of a phenomenon (GLMP) to model the assessment of integer arithmetic learning and implement the automated generation of a formative criterion-based assessment report in natural language, as well as a numerical grade. The report is generated based exclusively on the objective scores automatically generated by Moodle quizzes. GLMP is based on fuzzy logic and the computational theory of perceptions, and uses inference systems based on linguistic rules.

In a previous paper1, we proposed an axiomatic model for measuring self-contradiction in the framework of Atanassov fuzzy sets. This way, contradiction measures that are semicontinuous and completely semicontinuous, from both below and above, were defined. Although some examples were given, the problem of finding families of functions satisfying the different axioms remained open. The purpose of this paper is to construct some families of contradiction measures firstly using continuous t-norms and t-conorms, and secondly by means of strong negations. In both cases, we study the properties that they satisfy. These families are then classified according the different kinds of measures presented in the above paper.

We describe how to use a Granular Linguistic Model of a Phenomenon (GLMP) to assess e-learning processes. We apply this technique to evaluate algorithm learning using the GRAPHs learning environment.

The paper focuses on the study of the contradiction between two Atanassov's intuitionistic fuzzy sets. First, taking into account some characterizations obtained in previous papers, some functions are defined in order to measure the degrees of contradiction. Besides the principal properties of these measures are pointed out. Finally, some results relating self-contradiction and contradiction between two Atanassov's intuitionistic fuzzy sets are achieved.

In this poster, some tools for graph algorithms active learning are presented. These tools pursue a new goal on computer assisted educational resources, so as to acting as authentic virtual trainers extending teacher's hand through the Web. In other words, they are on line self assessment tools that help students to execute the algorithms by themselves, correcting their mistakes and providing students with clues to find the right solution. Furthermore, the tools might be used as complementary material for bLearning.