# Graphic Representation of a Dimensional Expansion of Triangular Fuzzy Number

Authors: Yong Sik Yun
Journal: Journal of Mathematics
 (2021)

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## Abstract

We calculate Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers. We prove that if the 3-dimensional result is limited to 2 dimensions, it is the same as the 2-dimensional result, which is shown as a graph. Since a 3-dimensional graph cannot be drawn, the value of the membership function is expressed with color density. We cut a 3-dimensional triangular fuzzy number by a perpendicular plane passing a vertex, and consider the cut plane as a domain. The value of the membership function for each point on the cut plane is also expressed with color density. The graph expressing the value of the membership function, defined in the plane as a 3-dimensional graph using thez-axis value instead of expressing with color density, is consistent with the results in the 2-dimensional case.


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## Body

## 1. Introduction

Many results exist for Zadeh’s max-min composition operators. The results for triangular fuzzy numbers are well known [1–5]. We have generalized 1-dimensional triangular fuzzy number to 2-dimensional triangular fuzzy number and calculated Zadeh’s max-min composition operators for 2-dimensional fuzzy numbers [6]. In the 1-dimensional case, Zadeh’s max-min composition operator can be calculated using α-cuts. By defining parametric operations between two region valued α-cuts, we obtained parametric operations for two triangular fuzzy numbers defined on ℝ2 [7]. In the case of 3-dimensional fuzzy numbers, the α-cuts are subsets of ℝ3. By defining parametric operations between two ellipsoids including interior valued α-cuts, we calculated Zadeh’s max-min composition operators for two 3-dimensional fuzzy numbers [8]. This will help facilitate further study of triangular fuzzy matrices [1, 3, 9].In this paper, we prove that Zadeh’s max-min composition operators for two 3-dimensional triangular fuzzy numbers onℝ3 constitutes the generalization of Zadeh’s max-min compositions for two 2-dimensional triangular fuzzy numbers on ℝ2. In addition, by limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result.


## 2. Preliminaries

We defineα-cut and α-set of the fuzzy set A on ℝ with the membership function μAx.Definition 1.
(see [10]). An α-cut of the fuzzy number A is defined by Aα=x∈ℝ|μAx≥α if α∈0,1 and A0=clx∈ℝ|μAx>α. 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.Following Zadeh, Dubois, and Prade, the extension principle is defined as follows.Definition 2.
(see [11–13]). The extended addition A+B, extended subtraction A−B, extended multiplication A⋅B, and extended division A/B are fuzzy sets with membership functions as follows. For all x∈A and y∈B,(1)μA∗Bz=supz=x∗yminμAx,μBy,∗=+,−,⋅,/.Definition 3.
(see [7]). A fuzzy set A with a membership function:(2)μAx,y=1−x−x12a2+y−y12b2,b2x−x12+a2y−y12≤a2b2,0,otherwise,where a,b>0 is called the 2-dimensional triangular fuzzy number and denoted by a,x1,b,y12.LetA=a,x1,b,y12. Then, the α-cut Aα of a 2-dimensional triangular fuzzy number A is an interior of an ellipse in an xy-plane, including the boundary(3)Aα=x,y∈ℝ2|x−x1a1−α2+y−y1b1−α2≤1.Theorem 1.
(see [7]). Let A 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.IfA is a continuous convex fuzzy number defined on ℝ2, then the α-cut Aα is a closed convex subset in ℝ2.Definition 4.
(see [7]). Let A and B be convex fuzzy numbers 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, we define that the parametric addition A+pB, parametric subtraction A−pB, parametric multiplication A⋅pB, and parametric division A/pB of two fuzzy numbers A and B are fuzzy numbers that have their α-sets as follows:(1)
A+pB: A+pBα=f1αt+g1αt,f2αt+g2αt∈ℝ2|0≤t≤2π.(2)
A−pB: A−pBα=xαt,yαt∈ℝ2|0≤t≤2π, where(6)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.(3)
A⋅pB: A⋅pBα=f1αt⋅g1αt,f2αt⋅g2αt∈ℝ2|0≤t≤2π.(4)
A/pB: A/pBα=xαt,yαt∈ℝ2|0≤t≤2π, where(7)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 [7]). Let A=a1,x1,b1,y12 and B=a2,x2,b2,y22 be two 2-dimensional triangular fuzzy numbers. Then, we have the following:(1)
A+pB=a1+a2,x1+x2,b1+b2,y1+y22.(2)
A−pB=a1+a2,x1−x2,b1+b2,y1−y22.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(8)xαt=x1x2+x1a2+x2a11−αcost+a1a21−α2cos2t,yαt=y1y2+y1b2+y2b11−αsint+b1b21−α2sin2t.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(9)xαt=x1+a11−αcostx2−a21−αcost,yαt=y1+b11−αsinty2−b21−αsint.Therefore,A+pB and A−pB become 2-dimensional triangular fuzzy numbers, but A⋅pB and A/pB are not 2-dimensional triangular fuzzy numbers.Theorem 3.
(see [7]). Parametric operations on ℝ2 in Definition 4 are the generalization of Zadeh’s extension principles on ℝ .


## 3. Zadeh’s Max-Min Composition Operator for 3-Dimensional Triangular Fuzzy Numbers

We define parametric operations between two 3-dimensional triangular fuzzy numbers and calculate Zadeh’s max-min composition operators for 3-dimensional triangular fuzzy numbers.Definition 5.
A fuzzy setA with a membership function:(10)μAx,y,z=1−x−x12a2+y−y12b2+z−z12c2,ifb2c2x−x12+c2a2y−y12+a2b2z−z12≤a2b2c2,0,otherwise,where a,b,c>0 is called the 3-dimensional triangular fuzzy number and denoted by 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 3-dimensional triangular fuzzy number A=a,x1,b,y1,c,z13 is the following set:(11)Aα=x,y,z∈ℝ3|x−x1a1−α2+y−y1b1−α2+z−z1c1−α2≤1.Definition 6.
A 3-dimensional fuzzy numberA defined on ℝ3 is called a convex fuzzy number if, for all α∈0,1, the α-cuts(12)Aα=x,y,z∈ℝ3|μAx,y,z≥α,are convex subsets in ℝ3.Theorem 4.
(see [8]). Let A 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(13)Aα=f1αs,f2αs,t,f3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.Definition 7.
LetA and B be two continuous convex fuzzy numbers defined on ℝ3 and(14)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-sets 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 numbers A and B are fuzzy numbers that have their α-sets as follows:(1)
Parametric addition,A+pB:(15)A+pBα=f1αs+g1αs,f2αs,t+g2αs,t,f3αs,t+g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.(2)
Parametric subtraction,A−pB:(16)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−π,f2αs,t−g2αs−π,t,f3αs,t−g3αs−π,t∈ℝ3|π≤s≤2π,−π2≤t≤π2.(3)
Parametric multiplication,A⋅pB:(17)A⋅pBα=f1αs⋅g1αs,f2αs,t⋅g2αs,t,f3αs,t⋅g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.(4)
Parametric division,A/pB:(18)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 5.
(see [8]). Let A=a1,x1,b1,y1,c1,z13 and B=a2,x2,b2,y2,c2,z23 be two 3-dimensional triangular fuzzy numbers. Then, we have the following:(1)
A+pB=a1+a2,x1+x2,b1+b2,y1+y2,c1+c2,z1+z23.(2)
A−pB=a1+a2,x1−x2,b1+b2,y1−y2,c1+c2,z1−z23.(3)
A⋅pBα=xαs,yαs,t,zαs,t∈ℝ3|0≤s≤2π,−π/2≤t≤π/2, where(19)xαs=x1x2+x1a2+x2a11−αcoss+a1a21−α2cos2s,yαs,t=y1y2+y1b2+y2b11−αsinscost+b1b21−α2sin2scos2t,zαs,t=z1z2+z1c2+z2c11−αsinssint+c1c21−α2sin2ssin2t.(4)
A/pBα=xαs,yαs,t,zαs,t∈ℝ3|0≤s≤2π,−π/2≤t≤π/2, where(20)xαs=x1+a11−αcossx2−a21−αcoss,yαs,t=y1+b11−αsinscosty2−b21−αsinscost,zαs,t=z1+c11−αsinssintz2−c21−αsinssint.Therefore,A+pB and A−pB become 3-dimensional triangular fuzzy numbers, but A⋅pB and A/pB are not 3-dimensional triangular fuzzy numbers.Theorem 6.
Parametric operations onℝ3 in Definition 7 are the generalization of parametric operations on ℝ2 in Definition 4, which are the generalization of Zadeh’s max-min composition operations on ℝ.Proof.
Consider two 3-dimensional triangular fuzzy numbersA=a1,x1,b1,y1,c1,03 and B=a2,x2,b2,y2,c2,03. By Theorem 5,(1)
A+pB=a1+a2,x1+x2,b1+b2,y1+y2,c1+c2,03.(2)
A−pB=a1+a2,x1−x2,b1+b2,y1−y2,c1+c2,03.(3)
A⋅pBα=xαs,yαs,t,zαs,t∈ℝ3|0≤s≤2π,−π/2≤t≤π/2, where(21)xαs=x1x2+x1a2+x2a11−αcoss+a1a21−α2cos2s,yαs,t=y1y2+y1b2+y2b11−αsinscost+b1b21−α2sin2scos2t,zαs,t=c1c21−α2sin2ssin2t.(4)
A/pBα=xαs,yαs,t,zαs,t∈ℝ3|0≤s≤2π,−π/2≤t≤π/2, where(22)xαs=x1+a11−αcossx2−a21−αcoss,yαs,t=y1+b11−αsinscosty2−b21−αsinscost,zαs,t=c11−αsinssint−c21−αsinssint=−c1c2.
The intersections of these 3-dimensional triangular fuzzy numbers andz=0 are as follows:(1)
A+pB: note that(23)μA+pBx,y,z=1−x−x1−x2a1+a22+y−y1−y2b1+b22+zc1+c22.Ifz=0,(24)μA+pBx,y,0=1−x−x1−x2a1+a22+y−y1−y2b1+b22.Thus, the intersection is the 2-dimensional triangular fuzzy numberC=a1+a2,x1+x2,b1+b2,y1+y22.(2)
A−pB: note that(25)μA−pBx,y,z=1−x−x1+x2a1+a22+y−y1+y2b1+b22+zc1+c22.Ifz=0,(26)μA−pBx,y,0=1−x−x1+x2a1+a22+y−y1+y2b1+b22.Thus, the intersection is the 2-dimensional triangular fuzzy numberD=a1+a2,x1−x2,b1+b2,y1−y22.(3)
A⋅pB: if z=0, c1c21−α2sin2ssin2t=0. Thus, the intersection is a fuzzy number E on ℝ2 with α-cut, not α-set, A⋅pBα=xαs,yαs,0∈ℝ2|0≤s≤2π, where(27)xαs=x1x2+x1a2+x2a11−αcoss+a1a21−α2cos2s,yαs,0=y1y2+y1b2+y2b11−αsinscos0+b1b21−α2sin2scos20=y1y2+y1b2+y2b11−αsins+b1b21−α2sin2s.(4)
A/pB: if z=0,−c1/c2=0. Thus, the intersection is a fuzzy number F on ℝ2 with α-cut, not α-set, A/pBα=xαs,yαs,0∈ℝ2|0≤s≤2π, where(28)xαs=x1+a11−αcossx2−a21−αcoss,yαs,0=y1+b11−αsinsy2−b21−αsins.
On the contrary, the intersection of 3-dimensional triangular fuzzy numberA=a1,x1,b1,y1,c1,03 and z=0 is G=a1,x1,b1,y12, and the intersection of 3-dimensional triangular fuzzy number B=a2,x2,b2,y2,c2,03 and z=0 is H=a2,x2,b2,y22. For two 2-dimensional triangular fuzzy numbers G and H, the following results for parametric operations on ℝ2 have been proven [7]:(29)G+pH=C,G−pH=D,G⋅pH=E,G/pH=F.
The proof is complete.


## 4. Conclusion

In this section, by limiting the graph of the 3-dimensional result to the 2-dimensional case, we prove that the result expressed as a graph is consistent with the graph of the 2-dimensional result.For two 3-dimensional triangular fuzzy numbersA=6,3,8,5,4,73 and B=4,6,6,8,3,53, the values of membership function are expressed using color density in Figures 1 and 2 , respectively. Zadeh’s max-min operations expanded to 3 dimensions are expressed in Figures 3–6 as graphs.Figure 1
A3.Figure 2
B3.Figure 3
A+B3.Figure 4
A−B3.Figure 5
A×B1/2.Figure 6
A/B1/2.The graph ofA⋅B and A/B for α=1/2 is Figures 5 and 6, respectively. In Figures 7 and 8 , the graph for A⋅B and A/B is the union of α0≤α≤1, respectively.Figure 7
A×B3.Figure 8
A/B3.We cut the graph ofA3,B3,A+B3, and A−B3 by a perpendicular plane passing a vertex (Figures 9–12 ). The values of the membership function of each point on the cross section are expressed with color density.Figure 9
A/2.Figure 10
B/2.Figure 11
A+B/2.Figure 12
A−B/2.We cut two 3-dimensional triangular fuzzy numbersA3 and B3 by a perpendicular plane passing a vertex and considered the cut plane as a domain. The value of the membership function for each point on the cut plane is expressed with color density. Figures 13 and 14 are the graphs expressing the value of the membership function defined in the plane as a 3-dimensional graph using the z-axis value, instead of expressing with color density. In this way, restricting the three-dimensional results (Figures 3–6) to two dimensions, we express the value of the membership function in a three-dimensional graph without using color density (Figures 15–18 ).Figure 13
A2.Figure 14
B2.Figure 15
A+B2.Figure 16
A−B2.Figure 17
A×B2.
(a)(b)Figure 18
A/B2.
(a)(b)Consider the 2-dimensional triangular fuzzy numbersA2 and B2 represented by Figures 13 and 14, respectively. The result of Zadeh’s max-min operators expanded to 2 dimensions can be found in [7]. When the results are graphed, it becomes clear that the graphs are identical to graphs in Figures 15–18. The commands in Mathematica for ten figures on page 7 and 8 are provided in appendix. Figure 19 summarizes what has been mentioned. In conclusion, therefore, the 3-dimensional result is naturally expanded to the 3 dimensions while satisfying the 2-dimensional result as it is, and it can be applied in this form. This paper will further study the extended applications of triangular fuzzy numbers [14–18].Figure 19
Summary.



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