# Three-Dimensional Expansion and Graphical Concept of Generalized Triangular Fuzzy Set

Authors: Yong Sik Yun
Journal: Journal of Mathematics
 (2022)

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## 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 inℝ2. And we generalized the triangular fuzzy numbers from ℝ2 to ℝ3. 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.


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## Body

## 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 ℝ2 [13–15]. In [16], we generalized the triangular fuzzy numbers from ℝ2 to ℝ3. By defining a parametric operator between two α-cuts with ellipsoidal values containing the interior, we defined a parametric operator for the two triangular fuzzy numbers defined in ℝ3. We proved that the results for the parametric operator are the generalization of Zadeh’s extended algebraic operator on ℝ [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 onℝ2

We defineα-cut and α-set of the fuzzy set A on ℝ with the membership function μAx.Definition 1.
Anα-cut of the fuzzy number A is defined by Aα=x∈ℝ∣μAx≥α if α∈0,1 and A0=clx∈ℝ∣μAx>α, where clB is the closure of B⊂ℝ. For α∈0,1, the set Aα=x∈X∣μAx=α is said to be the α-set of the fuzzy set A, A0 is the boundary of x∈ℝ∣μAx>α, and A1=A1.We define the generalized two-dimensional triangular fuzzy numbers onℝ2 as a generalization of generalized triangular fuzzy sets on ℝ and the parametric operations between two generalized two-dimensional triangular fuzzy sets. For that, we have to calculate operations between α-cuts in ℝ. The α-cuts are intervals in ℝ, but in ℝ2, the α-cuts are regions, which makes the existing method of calculations between α-cuts unusable. We interpret the existing method from a different perspective and apply the method to the region valued α-cuts on ℝ2.Definition 2.
A fuzzy setA with a membership function:(1)μAx,y=h−x−x12a2+y−y12b2,b2x−x12+a2y−y12≤a2b2h2,0,otherwise,where a,b>0 and 0<h<1 is called the generalized two-dimensional triangular fuzzy set and denoted by a,x1,h,b,y12.The intersections ofμAx,y and the vertical planes y−y1=kx−x1k∈ℝ are symmetric triangular fuzzy numbers in those planes. If a=b, ellipses become circles. The α-cut Aα of a generalized two-dimensional triangular fuzzy number A=a,x1,h,b,y12 is an interior of ellipse in an xy-plane including the boundary(2)Aα=x,y∈ℝ2|x−x1ah−α2+y−y1bh−α2≤1.Definition 3.
A two-dimensional fuzzy numberA defined on ℝ2 is called convex fuzzy number if for all α∈0,1, the α-cuts,(3)Aα=x,y∈ℝ2|μAx,y≥α,are convex subsets in ℝ2.Theorem 1 (see [9]).
LetA be a continuous convex fuzzy number defined on ℝ2 and Aα=x,y∈ℝ2|μAx,y=α be the α-set of A. Then, for all α∈0,1, there exist continuous functions f1αt and f2αt defined on 0,2π such that(4)Aα=f1αt,f2αt∈ℝ2|0≤t≤2π.Definition 5.
LetA and B be convex fuzzy sets defined on ℝ2 and(5)Aα=f1αt,f2αt∈ℝ2|0≤t≤2π,Bα=g1αt,g2αt∈ℝ2|0≤t≤2π,be the α-sets of A and B, respectively. For α∈0,1, the parametric addition, parametric subtraction, parametric multiplication, and parametric division are fuzzy sets that have their α-sets as follows.


### 2.1. Parametric AdditionA+pB

The parametric addition is given by the following:(6)A+pBα=f1αt+g1αt,f2αt+g2αt∈ℝ2|0≤t≤2π


### 2.2. Parametric SubtractionA−pB

The parametric subtraction is given by the following:(7)A−pBα=xαt,yαt∈ℝ2|0≤t≤2π,where(8)xαt=f1αt−g1αt+π,if0≤t≤π,f1αt−g1αt−π,ifπ≤t≤2π,yαt=f2αt−g2αt+π,if0≤t≤π,f2αt−g2αt−π,ifπ≤t≤2π.


### 2.3. Parametric MultiplicationA⋅pB

The parametric multiplication is given by the following:(9)A⋅pBα=f1αt⋅g1αt,f2αt⋅g2αt∈ℝ2|0≤t≤2π.


### 2.4. Parametric DivisionA/pB

The parametric division is given by the following:(10)A/pBα=xαt,yαt∈ℝ2|0≤t≤2π,where(11)xαt=f1αtg1αt+π,0≤t≤π,xαt=f1αtg1αt−π,π≤t≤2π,yαt=f2αtg2αt+π,0≤t≤π,yαt=f2αtg2αt−π,π≤t≤2π.Forα=0 and α=1, A∗pB0=limα⟶0+A∗pBα and A∗pB1=limα⟶1−A∗pBα, where ∗=+,−,⋅,/.Theorem 2 (see [10]).
LetA=a1,x1,h1,b1,y12 and B=a2,x2,h2,b2,y22 be two generalized two-dimensional triangular fuzzy sets. If 0<h1<h2<1, then we have the following:(1)
For0<α<h1, the α-set of A+pB is(12)A+pBα=x,y∈ℝ2|x−x1−x2a1h1−α+a2h2−α+y−y1−y2b1h1−α+b2h2−α2=1.(2)
For0<α<h1, the α-set of A−pB is(13)A−pBα=x,y∈ℝ2|x−x1+x2a1h1−α+a2h2−α2+y−y1+y2b1h1−α+b2h2−α2=1.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(14)xαt=x1x2+x1a2h2−α+x2a1h1−αcost+a1a2h1−αh2−αcos2t,0<α<h1,yαt=y1y2+y1b2h2−α+y2b1h1−αsint+b1b2h1−αh2−αsin2t,0<α<h1.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(15)xαt=x1+a1h1−αcostx2−a2h2−αcost,yαt=y1+b1h1−αsinty2−b2h2−αsint,0<α<h1.Furthermore, we have(16)A∗pB0=limα⟶0+A∗pBα,∗=+,−,⋅,/,A∗pBh1=limα⟶h1−A∗pBα,∗=+,−,⋅,/.Ifh1<α≤h2, by the Zadeh’s max-min principle operations, we obtain(17)A∗pBα=∅,∗=+,−,⋅,/.Example 1.
(see [10]). Let A=6,3,1/2,8,52 and B=4,2,2/3,5,32. Then, by Theorem 2, we have the following:(1)
For0<α<1/2, the α-set of A+pB is(18)A+pBα=x,y∈ℝ2|3x−1517−30α2+3y−2422−39α2=1.(2)
For0<α<1/2, the α-set of A−pB is(19)A−pBα=x,y∈ℝ2|3x−317−30α2+3y−622−39α2=1.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(20)xαt=6+14−24αcost+41−2α2−3αcos2t,0<α<12,yαt=15+863−49αsint+201−2α23−αsin2t,0<α<12.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(21)xαt=9+91−2αcost6−42−3αcost,yαt=15+121−2αsint9−152−3αsint,0<α<12.


## 2.1. Parametric AdditionA+pB

The parametric addition is given by the following:(6)A+pBα=f1αt+g1αt,f2αt+g2αt∈ℝ2|0≤t≤2π


## 2.2. Parametric SubtractionA−pB

The parametric subtraction is given by the following:(7)A−pBα=xαt,yαt∈ℝ2|0≤t≤2π,where(8)xαt=f1αt−g1αt+π,if0≤t≤π,f1αt−g1αt−π,ifπ≤t≤2π,yαt=f2αt−g2αt+π,if0≤t≤π,f2αt−g2αt−π,ifπ≤t≤2π.


## 2.3. Parametric MultiplicationA⋅pB

The parametric multiplication is given by the following:(9)A⋅pBα=f1αt⋅g1αt,f2αt⋅g2αt∈ℝ2|0≤t≤2π.


## 2.4. Parametric DivisionA/pB

The parametric division is given by the following:(10)A/pBα=xαt,yαt∈ℝ2|0≤t≤2π,where(11)xαt=f1αtg1αt+π,0≤t≤π,xαt=f1αtg1αt−π,π≤t≤2π,yαt=f2αtg2αt+π,0≤t≤π,yαt=f2αtg2αt−π,π≤t≤2π.Forα=0 and α=1, A∗pB0=limα⟶0+A∗pBα and A∗pB1=limα⟶1−A∗pBα, where ∗=+,−,⋅,/.Theorem 2 (see [10]).
LetA=a1,x1,h1,b1,y12 and B=a2,x2,h2,b2,y22 be two generalized two-dimensional triangular fuzzy sets. If 0<h1<h2<1, then we have the following:(1)
For0<α<h1, the α-set of A+pB is(12)A+pBα=x,y∈ℝ2|x−x1−x2a1h1−α+a2h2−α+y−y1−y2b1h1−α+b2h2−α2=1.(2)
For0<α<h1, the α-set of A−pB is(13)A−pBα=x,y∈ℝ2|x−x1+x2a1h1−α+a2h2−α2+y−y1+y2b1h1−α+b2h2−α2=1.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(14)xαt=x1x2+x1a2h2−α+x2a1h1−αcost+a1a2h1−αh2−αcos2t,0<α<h1,yαt=y1y2+y1b2h2−α+y2b1h1−αsint+b1b2h1−αh2−αsin2t,0<α<h1.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(15)xαt=x1+a1h1−αcostx2−a2h2−αcost,yαt=y1+b1h1−αsinty2−b2h2−αsint,0<α<h1.Furthermore, we have(16)A∗pB0=limα⟶0+A∗pBα,∗=+,−,⋅,/,A∗pBh1=limα⟶h1−A∗pBα,∗=+,−,⋅,/.Ifh1<α≤h2, by the Zadeh’s max-min principle operations, we obtain(17)A∗pBα=∅,∗=+,−,⋅,/.Example 1.
(see [10]). Let A=6,3,1/2,8,52 and B=4,2,2/3,5,32. Then, by Theorem 2, we have the following:(1)
For0<α<1/2, the α-set of A+pB is(18)A+pBα=x,y∈ℝ2|3x−1517−30α2+3y−2422−39α2=1.(2)
For0<α<1/2, the α-set of A−pB is(19)A−pBα=x,y∈ℝ2|3x−317−30α2+3y−622−39α2=1.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(20)xαt=6+14−24αcost+41−2α2−3αcos2t,0<α<12,yαt=15+863−49αsint+201−2α23−αsin2t,0<α<12.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(21)xαt=9+91−2αcost6−42−3αcost,yαt=15+121−2αsint9−152−3αsint,0<α<12.


## 3. Parametric Operations for Generalized Three-Dimensional Triangular Fuzzy Sets onℝ3

We define the generalized three-dimensional triangular fuzzy sets onℝ3 as a generalization of generalized triangular fuzzy sets on ℝ2. Then, we define the parametric operations between two generalized three-dimensional triangular fuzzy sets. For that, we have to calculate operations between α-sets in ℝ3. The α-sets are regions in ℝ2, but in ℝ3, the α-sets are ellipsoids including interior, which makes the existing method of calculations between α-sets unusable. We interpret the existing method from a different perspective and apply the method to the ellipsoids including interior-valued α-sets on ℝ3.Definition 6.
A fuzzy setA with a membership function μAx,y,z such that(22)h−x−x12a2+y−y12b2+z−z12c2,ifb2c2x−x12+c2a2y−y12+a2b2z−z12≤a2b2c2h2,0,otherwise,where a,b,c>0 and 0<h<1 is called the generalized three-dimensional triangular fuzzy set and denoted by h,a,x1,b,y1,c,z13.Note thatμAx,y is a cone in ℝ2, but we cannot know the shape of μAx,y,z in ℝ3. The α-cut Aα of a generalized three-dimensional triangular fuzzy number A=h,a,x1,b,y1,c,z13 is the following set:(23)Aα=x,y,z∈ℝ3|x−x1ah−α2+y−y1bh−α2+z−z1ch−α2≤1.Definition 7.
A three-dimensional fuzzy numberA defined on ℝ3 is called convex fuzzy number if for all α∈0,1, the α-cuts,(24)Aα=x,y,z∈ℝ3|μAx,y,z≥α,are convex subsets in ℝ3.Theorem 3 (see [17]).
LetA be a continuous convex fuzzy number defined on ℝ3 and Aα=x,y,z∈ℝ3|μAx,y,z=α be the α-set of A. Then, for all α∈0,1, there exist continuous functions f1αs,f2αs,t, and f3αs,t0≤s≤2π,−π/2≤t≤π/2 such that(25)Aα=f1αs,f2αs,t,f3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.Definition 8 (see [17]).
LetA and B are two continuous convex fuzzy sets defined on ℝ3 and(26)Aα=f1αs,f2αs,t,f3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2,Bα=g1αs,g2αs,t,g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2,be the α-set of A and B, respectively. For α∈0,1, we define that the parametric addition, parametric subtraction, parametric multiplication, and parametric division of two fuzzy sets A and B are fuzzy numbers that have their α-sets as follows:(1)
Parametric additionA+pB:(27)A+pBα=f1αs+g1αs,f2αs,t+g2αs,t,f3αs,t+g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.(2)
Parametric subtractionA−pB:(28)A−pBα=f1αs−g1αs+π,f2αs,t−g2αs+π,t,f3αs,t−g3αs+π,t∈ℝ3|0≤s≤π,−π2≤t≤π2,A−pBα=f1αs−g1αs−πf2αs,t−g2αs−π,t,f3αs,t−g3αs−π,t∈ℝ3|π≤s≤2π,−π2≤t≤π2.(3)
Parametric multiplicationA⋅pB:(29)A⋅pBα=f1αs⋅g1αs,f2αs,t⋅g2αs,t,f3αs,t⋅g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.(4)
Parametric divisionA/pB:(30)A/pBα=f1αsg1αs+π,f2αs,tg2αs+π,t,f3αs,tg3αs+π,t∈ℝ3|0≤s≤π,−π2≤t≤π2,A/pBα=f1αsg1αs−π,f2αs,tg2αs−π,t,f3αs,tg3αs−π,t∈ℝ3|π≤s≤2π,−π2≤t≤π2.
Forα=0 and α=1, A∗pB0=limα⟶0+A∗pBα and A∗pB1=limα⟶1−A∗pBα, where ∗=+,−,⋅,/.Theorem 4 (see [17]).
LetA=h1,a1,x1,b1,y1,c1,z13 and B=h2,a2,x2,b2,y2,c2,z23 be two generalized three-dimensional triangular fuzzy sets. If 0<h1<h2<1, then we have the following:(1)
For0<α<h1, the α-set of A+pB is(31)A+pBα=x,y,z∈ℝ3|x−x1−x2a1h1−α+a2h2−α2+y−y1−y2b1h1−α+b2h2−α2+z−z1−z2c1h1−α+c2h2−α2=1.(2)
For0<α<h1, the α-set of A−pB is(32)A−pBα=x,y,z∈ℝ3|x−x1+x2a1h1−α+a2h2−α2+y−y1+y2b1h1−α+b2h2−α2+z−z1+z2c1h1−α+c2h2−α2=1.(3)
For0<α<h1, A⋅pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2, where(33)xαs=x1x2+x1a2h2−α+x2a1h1−αcoss+a1a2h1−αh2−αcos2s,yαs,t=y1y2+y1b2h2−α+y2b1h1−αsinscost+b1b2h1−αh2−αsin2scos2t,zαs,t=z1z2+z1c2h2−α+z2c1h1−αsinssint+c1c2h1−αh2−αsin2ssin2t.(4)
For0<α<h1, A/pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2, where(34)xαs=x1+a1h1−αcossx2−a2h2−αcoss,yαs.t=y1+b1h1−αsinscosty2−b2h2−αsinscost,zαs.t=z1+c1h1−αsinssintz2−c2h2−αsinssint.
Furthermore, we have(35)A∗pB0=limα⟶0+A∗pBα,∗=+,−,⋅,/,A∗pBh1=limα⟶h1−A∗pBα,∗=+,−,⋅,/,
Ifh1<α≤h2, by the Zadeh’s max-min principle operations, we obtain(36)A∗pBα=∅,∗=+,−,⋅,/,


## 4. A Generalization fromℝ2 to ℝ3 of Generalized Triangular Fuzzy Sets

In this section, we show that the parametric operations for two generalized triangular fuzzy sets defined onℝ3 are a generalization of parametric operations for two generalized triangular fuzzy sets defined on ℝ2. For that, we have to prove that the intersections of the results on ℝ3 and z=0 are the same as those on ℝ2.Theorem 5.
For∗=+,−,⋅,/, let μA∗Bx,y,z and μA∗Bx,y are the results in Theorem 3 and Theorem 2, respectively. Then, we have μA∗Bx,y,0=μA∗Bx,y.Proof.
ConsiderA=h1,a1,x1,b1,y1,0,03 and B=h2,a2,x2,b2,y2,0,03, where 0<h1<h2<1.(1)
For0<α<h1, the α-set of μA+Bx,y,0 is(37)x,y∈ℝ2|x−x1−x2a1h1−α+a2h2−α2+y−y1−y2b1h1−α+b2h2−α2=1.Similarly, we can prove that the0−set and h1−set of μA+Bx,y,0 are the same as those of μA+Bx,y.(2)
For0<α<h1, the α-set of μA−Bx,y,0 is(38)x,y∈ℝ2|x−x1+x2a1h1−α+a2h2−α2+y−y1+y2b1h1−α+b2h2−α2=1.Similarly, we can prove that the0−set and h1−set of μA−Bx,y,0 are the same as those of μA−Bx,y.(3)
For0<α<h1, the α-set of μA⋅Bx,y,0 is(39)S1=xαs,yαs,t∈ℝ2|0≤s≤2π,−π2≤t≤π2,where(40)xαs=x1x2+x1a2h2−α+x2a1h1−αcoss+a1a2h1−αh2−αcos2s,yαs,t=y1y2+y1b2h2−α+y2b1h1−αsinscost+b1b2h1−αh2−αsin2scos2t.In Theorem2, the α-set of μA⋅Bx,y is(41)S2=xαt,yαt∈ℝ2|0≤t≤2π,where(42)xαt=x1x2+x1a2h2−α+x2a1h1−αcost+a1a2h1−αh2−αcos2t,yαt=y1y2+y1b2h2−α+y2b1h1−αsint+b1b2h1−αh2−αsin2t.In three-dimensional case, theα−set becomes a convex set in ℝ2. The boundary of S1 is S2. Clearly, x0s=x0t,xh1s=yh1t, and we can prove that(43)y0s,t=y0t,yh1s,t=yh1t.(4)
For0<α<h1, the α-set of μA/Bx,y,0 is(44)S3=xαs,yαs,t∈ℝ2|0≤s≤2π,−π2≤t≤π2,where(45)xαs=x1+a1h1−αcossx2−a2h2−αcoss,yαs.t=y1+b1h1−αsinscosty2−b2h2−αsinscost.
In Theorem2, the α-set of μA/Bx,y is(46)S4=xαt,yαt∈ℝ2|0≤t≤2π,where(47)xαt=x1+a1h1−αcostx2−a2h2−αcost,yαt=y1+b1h1−αsinty2−b2h2−αsint.
In a three-dimensional case, theα-set becomes a convex set in ℝ2. The boundary of S3 is S4. Clearly, x0s=x0t,xh1s=yh1t, and we can prove that(48)y0s,t=y0t,yh1s,t=yh1t.
Thus,μA∗Bx,y,0=μA∗Bx,y.


## 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 (Figures1–6). 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).Figure 1
A2D.Figure 2
B2D.Figure 3
A3D.Figure 4
B3D.Figure 5
A3Dhalf−cut.Figure 6
B3Dhalf−cut.Figure 7
A+B2Da−cut.Figure 8
A−B2Da−cut.Figure 9
AB2Da−cut.Figure 10
AdB2Da−cut.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 (Figures11–14).Figure 11
A+B3Dhalf−cut.Figure 12
A−B3Dhalf−cut.Figure 13
AB3Dhalf−cut.Figure 14
AdB3Dhalf−cut.Each point on the cross section has a different color density, noting that each has a specific function value. In Section4, 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.Figure 15
A+B2D=3D.Figure 16
A−B2D=3D.Figure 17
AB2D=3D.Figure 18
AdB2D=3D.In Section1, 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 ℝ2 and generalized the triangular fuzzy numbers from ℝ2 to ℝ3. The purpose of the paper is also presented.In Section2, we defined the generalized two-dimensional triangular fuzzy numbers on ℝ2 as a generalization of generalized triangular fuzzy sets on ℝ 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 Section3, we defined the generalized three-dimensional triangular fuzzy sets on ℝ3 as a generalization of generalized triangular fuzzy sets on ℝ2. 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 Section4, we showed that the parametric operations for two generalized triangular fuzzy sets defined on ℝ3 are a generalization of parametric operations for two generalized triangular fuzzy sets defined on ℝ2. What has been proven is presented as an example. And the examples are expressed in various types of graphs for easier understanding.



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*Source: 3248127-2022-04-26.xml*