Abstract
We have studied the extended algebraic operations between two fuzzy numbers and calculated Zadeh’s max-min composition operator for two generalized triangular fuzzy sets in . And we generalized the triangular fuzzy numbers from to . We prove that the result of the three-dimensional case is an extension of two-dimensional case and presented it in a graph. The extension is proved by showing that the result obtained by restricting the three-dimensional result to two-dimensional result is consistent with the existing two-dimensional result.
1. Introduction
Fuzzy theory has been increasingly applied to humanities including logics and sociology as well as natural sciences from engineering to medicine. In mathematics, triangular fuzzy sets have been extensively studied, which resulted in numerous fuzzy theories. In applications of the fuzzy set theories, many operators between two fuzzy sets have been defined and calculated. In particular, Zadeh’s operators have been widely applied and developed [1–3]. Recently, the application expands to fuzzy control theory [4, 5] and fuzzy logic [6–8]. The theories of triangular fuzzy numbers have been extended to generalized triangular fuzzy sets that do not have the maximum value of 1. And as the number of ambiguous fuzzy variables increases, the theories have been extended to the studies of two-dimensional and three-dimensional fuzzy sets. In that respect, the study that extended Zadeh’s operator theory to two or three dimensions is meaningful. We have studied the extended algebraic operations between two fuzzy numbers [9–12] and calculated Zadeh’s max-min composition operator for two generalized triangular fuzzy sets in [13–15]. In [16], we generalized the triangular fuzzy numbers from to . By defining a parametric operator between two -cuts with ellipsoidal values containing the interior, we defined a parametric operator for the two triangular fuzzy numbers defined in . We proved that the results for the parametric operator are the generalization of Zadeh’s extended algebraic operator on [9]. In addition, we calculated the parametric operators for two generalized three-dimensional triangular fuzzy sets and presented the calculation in three-dimensional graphs [17].
In this paper, we prove that the result of the three-dimensional case is an extension of two dimensions and presented it in a graph. The extension is proved by showing that the result obtained by restricting the three-dimensional result to two dimensions is consistent with the existing two-dimensional result. The graph of the fuzzy set defined in three dimensions expresses the function value by color density. When the graph is cut with a vertical plane passing through the vertex of a generalized three-dimensional triangular fuzzy set, the function value is shown through color density on the cross section of the graph. The value of the membership function defined on the cross section can be expressed in a graph of the function defined in two dimensions. We show that this graph is consistent with the three-dimensional representation of the results in two dimensions.
2. Zadeh’s Max-Min Composition Operations for Generalized Triangular Fuzzy Sets on
We define -cut and -set of the fuzzy set on with the membership function .
Definition 1. An -cut of the fuzzy number is defined by if and , where is the closure of . For , the set is said to be the -set of the fuzzy set , is the boundary of , and .
We define the generalized two-dimensional triangular fuzzy numbers on as a generalization of generalized triangular fuzzy sets on and the parametric operations between two generalized two-dimensional triangular fuzzy sets. For that, we have to calculate operations between -cuts in . The -cuts are intervals in , but in , the -cuts are regions, which makes the existing method of calculations between -cuts unusable. We interpret the existing method from a different perspective and apply the method to the region valued -cuts on .
Definition 2. A fuzzy set with a membership function:where and is called the generalized two-dimensional triangular fuzzy set and denoted by .
The intersections of and the vertical planes are symmetric triangular fuzzy numbers in those planes. If , ellipses become circles. The -cut of a generalized two-dimensional triangular fuzzy number is an interior of ellipse in an -plane including the boundary
Definition 3. A two-dimensional fuzzy number defined on is called convex fuzzy number if for all , the -cuts,are convex subsets in .
Theorem 1 (see [9]). Let be a continuous convex fuzzy number defined on and be the -set of . Then, for all , there exist continuous functions and defined on such that
Definition 5. Let and be convex fuzzy sets defined on andbe the -sets of and , respectively. For , the parametric addition, parametric subtraction, parametric multiplication, and parametric division are fuzzy sets that have their -sets as follows.
2.1. Parametric Addition
The parametric addition is given by the following:
2.2. Parametric Subtraction
The parametric subtraction is given by the following:where
2.3. Parametric Multiplication
The parametric multiplication is given by the following:
2.4. Parametric Division
The parametric division is given by the following:where
For and , and , where .
Theorem 2 (see [10]). Let and be two generalized two-dimensional triangular fuzzy sets. If , then we have the following:(1)For , the -set of is(2)For , the -set of is(3), where(4), where
Furthermore, we have
If , by the Zadeh’s max-min principle operations, we obtain
Example 1. (see [10]). Let and . Then, by Theorem 2, we have the following:(1)For , the -set of is(2)For , the -set of is(3), where(4), where
3. Parametric Operations for Generalized Three-Dimensional Triangular Fuzzy Sets on
We define the generalized three-dimensional triangular fuzzy sets on as a generalization of generalized triangular fuzzy sets on . Then, we define the parametric operations between two generalized three-dimensional triangular fuzzy sets. For that, we have to calculate operations between -sets in . The -sets are regions in , but in , the -sets are ellipsoids including interior, which makes the existing method of calculations between -sets unusable. We interpret the existing method from a different perspective and apply the method to the ellipsoids including interior-valued -sets on .
Definition 6. A fuzzy set with a membership function such thatwhere and is called the generalized three-dimensional triangular fuzzy set and denoted by .
Note that is a cone in , but we cannot know the shape of in . The -cut of a generalized three-dimensional triangular fuzzy number is the following set:
Definition 7. A three-dimensional fuzzy number defined on is called convex fuzzy number if for all , the -cuts,are convex subsets in .
Theorem 3 (see [17]). Let be a continuous convex fuzzy number defined on and be the -set of . Then, for all , there exist continuous functions , and such that
Definition 8 (see [17]). Let and are two continuous convex fuzzy sets defined on andbe the -set of and , respectively. For , we define that the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy sets and are fuzzy numbers that have their -sets as follows:(1)Parametric addition :(2)Parametric subtraction :(3)Parametric multiplication :(4)Parametric division :For and , and , where .
Theorem 4 (see [17]). Let and be two generalized three-dimensional triangular fuzzy sets. If , then we have the following:(1)For , the -set of is(2)For , the -set of is(3)For , , where(4)For , , whereFurthermore, we haveIf , by the Zadeh’s max-min principle operations, we obtain
4. A Generalization from to of Generalized Triangular Fuzzy Sets
In this section, we show that the parametric operations for two generalized triangular fuzzy sets defined on are a generalization of parametric operations for two generalized triangular fuzzy sets defined on . For that, we have to prove that the intersections of the results on and are the same as those on .
Theorem 5. For , let and are the results in Theorem 3 and Theorem 2, respectively. Then, we have .
Proof. Consider and , where .(1)For , the -set of is Similarly, we can prove that the set and set of are the same as those of .(2)For , the -set of is Similarly, we can prove that the set and set of are the same as those of .(3)For , the -set of is where In Theorem 2, the -set of is where In three-dimensional case, the set becomes a convex set in . The boundary of is . Clearly, , and we can prove that(4)For , the -set of iswhereIn Theorem 2, the -set of iswhereIn a three-dimensional case, the -set becomes a convex set in . The boundary of is . Clearly, , and we can prove thatThus, .
5. Conclusion
Extensive use of fuzzy theory in many different fields has facilitated active research on operators between fuzzy sets [18–20]. Operators of various concepts have been defined and studied, but Zadeh’s operator concept is commonly studied and utilized. A correct understanding of the generalized triangular fuzzy set will be helpful in interpreting Zadeh’s operators [21–24].
The conclusion of a general triangular fuzzy set in two-dimensional space is a function defined on a plane and can be expressed in a graph in three-dimensional space (Figures 1–6). Therefore, the graph in the form of an elliptical cone in two dimensions, the result of Section 2, is not difficult to understand (Figures 7–10).
However, if it is expanded to three dimensions, the domain becomes a three-dimensional set, making visual expression difficult. To help easier understanding, graphs were expressed in color density (Figures 11–14).
Each point on the cross section has a different color density, noting that each has a specific function value. In Section 4, we proved that the three-dimensional result is an extended concept of the two-dimensional result, which is indicated in graphs in Figures 15–18. When the three-dimensional result is cut with a vertical plane passing through the vertex, the cross section of the graph is two-dimensional. Function values in two-dimensional were represented in graphs with the z-axis value. Visualization of the results will lead to more application and utilization.
In Section 1, we discussed the theoretical flow and application of fuzzy sets. The study that extended Zadeh’s operator theory to two or three dimensions is important in application. We have studied the extended algebraic operations between two fuzzy numbers and calculated Zadeh’s max-min composition operator for two generalized triangular fuzzy sets in and generalized the triangular fuzzy numbers from to . The purpose of the paper is also presented.
In Section 2, we defined the generalized two-dimensional triangular fuzzy numbers on as a generalization of generalized triangular fuzzy sets on and the parametric operations between two generalized two-dimensional triangular fuzzy sets. In Theorem 2, we calculated the parametric operations between two generalized two-dimensional triangular fuzzy sets and gave an example.
In Section 3, we defined the generalized three-dimensional triangular fuzzy sets on as a generalization of generalized triangular fuzzy sets on . Then, we defined the parametric operations between two generalized three-dimensional triangular fuzzy sets. We calculated the parametric operations between two generalized three-dimensional triangular fuzzy sets in Theorem 3.
In Section 4, we showed that the parametric operations for two generalized triangular fuzzy sets defined on are a generalization of parametric operations for two generalized triangular fuzzy sets defined on . What has been proven is presented as an example. And the examples are expressed in various types of graphs for easier understanding.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this article.
Acknowledgments
This research was supported by the 2022 Scientific Promotion Program funded by Jeju National University. Some of the 2D results have been published previously.