Symmetric difference operators on fuzzy sets☆
Introduction
Given a set X, the symmetric difference operator on the powerset refers to the binary operator given by The operator △ has the following properties: for all ,
- (a)
, (symmetry)
- (b)
,
- (c)
,
- (d)
, (associativity)
- (e)
.
Symmetric difference is a basic operation on sets. So, since the introduction of fuzzy sets by Zadeh [26], it has been an interesting topic to extend this operation to the fuzzy setting, see e.g. [1], [7], [9], [10], [14], [15], [18], [20], [22].
The symmetric difference operator on the powerset is clearly determined by the function given by Analogously, a symmetric difference operator on the fuzzy powerset reduces to an operator that satisfies certain conditions.
There are two approaches to the study of symmetric difference operators on fuzzy sets. The first is axiomatic, the second is model theoretic. Both approaches are based, of course, on the symmetric difference between crisp sets.
The axiomatic approach starts from the properties (a)–(e) of the symmetric difference between crisp sets. It would be desirable if one could find functions with the following conditions: for ,
- (i)
, (symmetry)
- (ii)
,
- (iii)
,
- (iv)
, (associativity)
- (v)
,
In [7], dropping associativity, Alsina and Trillas defined a symmetric difference operator to be a function that satisfies the conditions (ii), (iii), and (v). In 2013, Dombi [14] investigated the properties of these operators, and proposed a new definition of symmetric difference operators by modifying (v). The articles [10], [20] are also concerned with such operators.
In [9], Bedregal, Reiser, and Dimuro defined a symmetric difference operator to be a function that satisfies (i), (ii), (iv), and the boundary condition (a weakening of (v)). The articles [18], [20] continued the study of these operators.
The model theoretic approach is based on the following formulas for symmetric difference between crisp sets: for any , The analogue of these formulas in fuzzy set theory are respectively and where, S is a continuous t-conorm, T is a continuous t-norm, and n is a strong negation on . It is clear that functions of the form or satisfy the conditions (i), (ii), (iii), and the boundary condition . So, they have been studied as natural candidates for symmetric difference operators in fuzzy set theory, see e.g. [1], [3], [4], [9], [19]. Then, it is natural to ask: when are they associative? The reader is referred to [3], [4], [9], [18], [20] for more on this topic. Relevant works can also be found in [2], [5], [6], [8], [11], [12], [13].
In this article, being aware of the importance of the associativity of a binary operator, we are concerned with symmetric difference operators that satisfy the associative law. But, unlike [9], we define a symmetric difference operator to be a continuous function that satisfies (ii), (iv), and the boundary condition . As shall be seen, such operators have many nice properties. For instance, it will be shown that they also satisfy the conditions (i) and (iii). A systematic investigation of the properties and structures of these operators is presented here. The main results are: (1) It is proved that such an operator is determined by a continuous t-conorm and a strong negation. (2) Some conditions are obtained for functions of the form or to be associative (hence symmetric difference operators). These results are related to the solution of certain functional equations on the unit interval . In particular, the results presented here provide a partial answer to a question raised by Alsina, Frank, and Schweizer.
The contents are arranged as follows. Section 2 recalls basic properties of continuous t-norms, continuous t-conorms, and strong negations that will be needed in subsequent sections. Section 3 is devoted to the structure and properties of symmetric difference operators. Section 4 and Section 5 are concerned with two models of these operators. Section 6 contains concluding remarks and a question.
Section snippets
Continuous t-norms and continuous t-conorms
Triangular norms (t-norms, for short) were introduced by Schweizer and Sklar [23], [24] in the study of probabilistic metric spaces as a special kind of associative functions defined on the unit interval. These functions have found applications in many areas since then. In particular, continuous t-norms and their dual, continuous t-conorms, play a prominent role in fuzzy set theory [17], [19]. The monographs [4], [19] are our main references to t-norms and t-conorms. We recall here some basic
Symmetric difference operators
Definition 3.1 A two-place function is called a symmetric difference operator if the following conditions are satisfied: for all x in , , △ is associative, i.e. for all in , △ is continuous.
Proposition 3.2 If △ is a symmetric difference operator, then for all and is a strong negation on . Proof Let . Then showing that . So, f is a strong negation since f is
The model
Suppose T is a continuous t-norm, S is a continuous t-conorm and n is a strong negation. Then the function , given by is continuous and satisfies that for all in ,
- (i)
;
- (ii)
;
- (iii)
.
The model
Suppose T is a continuous t-norm, S is a continuous t-conorm and n is a strong negation. Then the function , given by is continuous and satisfies that for all in :
- (i)
;
- (ii)
;
- (iii)
.
Concluding remarks
The study of operators on fuzzy sets, for instance, complement, difference, and symmetric difference, often reduces to a study of certain binary operators on the unit interval. In this article, a symmetric difference operator (on fuzzy sets) is defined to be a continuous function that is associative and satisfies the boundary condition and . The structure of such operators has been determined and two models have been investigated. In particular, the
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