# An Expanded Concept and Graph of Zadeh’s Algebraic Operations for 3-Dimensional Generalized Triangular Fuzzy Sets

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

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## Abstract

Algebraic operators for fuzzy sets have been studied in 2-dimensional space. We define the algebraic operators between two 3-dimensional fuzzy sets and calculate the operators for the generalized 3-dimensional triangular fuzzy sets. To facilitate an understanding of the results in 3D, the values of membership function are displayed in graphs using color density. We give various types of graphs ofα-cuts.


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## Body

## 1. Introduction

Many concepts defined by Zadeh have been developed in various fields of fuzzy theory and have been recently applied to fuzzy logic and fuzzy control [1–3]. The multivalued logic theorem was applied to fuzzy logic, and fuzzy operator theory was applied to fuzzy control [2, 4]. Theories on algebraic operators have been established by scholars, and studies on corresponding operators have been actively conducted [5–7]. One of the efficient ways to understand various operators is to apply them to a common, simple triangular fuzzy number [8–11].With the number of uncertain factors increasing, fuzzy theories in 2-dimensional space have been researched [12–14]. Algebraic operators in 2-dimensional space are applied to many fields [15, 16]. When the results of the 3-dimensional algebraic operators on triangular fuzzy numbers are restricted to 2D, the results are consistent with those of 1D. Since the cases where the maximum value is not 1 are commonly found, the theories of general fuzzy sets can be more widely applied. Numerous studies on 2-dimensional triangular fuzzy sets which do not have the maximum value of 1 have been conducted [17–19].In this paper, we define the algebraic operators between two 3-dimensional fuzzy sets and calculate the operators for the generalized 3-dimensional triangular fuzzy sets. To facilitate an understanding of the results in 3D, the values of membership function are displayed in graphs using color density. Various types of the graphs in this paper allow us to interpret the shape of the alpha cut, which may help promote further applications.


## 2. Preliminaries

We defineα-cut and α-set of the fuzzy set A on ℝ with the membership function μAx.Definition 1 (see [20]).
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. 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 [21–23]).
The extended additionA+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.
A fuzzy setA with a membership function(2)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 2-dimensional triangular fuzzy set and denoted by a,x1,h,b,y12.Theorem 1 (see [24]).
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(3)Aα=f1αt,f2αt∈ℝ2|0≤t≤2π.We defined the parametric addition, parametric subtraction, parametric multiplication, and parametric division for two convex fuzzy setsA and B defined on ℝ2 [25].Theorem 2 (see [25]).
LetA=a1,x1,h1,b1,y12 and B=a2,x2,h2,b2,y22 be two generalized 2-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(4)A+pBα=x,y∈ℝ2|x−x1−x2a1h1−α+a2h2−α2+y−y1−y2b1h1−α+b2h2−α2=1.Furthermore, we have(5)A+pB0=x,y∈ℝ2|x−x1−x2a1h1+a2h22+y−y1−y2b1h1+b2h22=1,A+pBh1=x,y∈ℝ2|x−x1−x2a2h2−h12+y−y1−y2b2h2−h12=1,A+pBα=∅,h1<α≤h2.(2)
For0<α<h1, the α-set of A−pB is(6)A−pBα=x,y∈ℝ2|x−x1+x2a1h1−α+a2h2−α2+y−y1+y2b1h1−α+b2h2−α2=1.Furthermore, we have(7)A−pB0=x,y∈ℝ2|x−x1+x2a1h1+a2h22+y−y1+y2b1h1+b2h22=1,A−pBh1=x,y∈ℝ2|x−x1+x2a2h2−h12+y−y1+y2b2h2−h12=1,A−pBα=∅,h1<α≤h2.(3)
A⋅pBα=xαt,yαt|0≤t≤2π, where(8)xαt=x1x2+x1a2h2−α+x2a1h1−αcost+a1a2h1−αh2−αcos2t,0<α<h1,yαt=y1y2+y1b2h2−α+y2b1h1−αsint+b1b2h1−αh2−αsin2t,0<α<h1.Furthermore, we have(9)x0t=x1x2+x1a2h2+x2a1h1cost+a1a2h1h2cos2t,xh1t=x1x2+x1a2h2−h1cost,y0t=y1y2+y1b2h2+y2b1h1sint+b1b2h1h2sin2t,yh1t=y1y2+y1b2h2−h1sint,A⋅pBα=∅,h1<α≤h2.(4)
A/pBα=xαt,yαt|0≤t≤2π, where(10)xαt=x1+a1h1−αcostx2−a2h2−αcost,yαt=y1+b1h1−αsinty2−b2h2−αsint,0<α<h1.
Furthermore, we have(11)x0t=x1+a1h1costx2−a2h2cost,xh1t=x1x2−a2h2−h1cost,y0t=y1+b1h1sinty2−b2h2sint,yh1t=y1y2−b2h2−h1sint,A/pBα=∅,h1<α≤h2.


## 3. A Generalized 3-Dimensional Triangular Fuzzy Set

Definition 4.
A fuzzy setA with a membership function(12)μAx,y,z=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 3-dimensional triangular fuzzy set and denoted by h,a,x1,b,y1,c,z13.
For0<α<h, the α-cut Aα of a generalized 3-dimensional triangular fuzzy set A=h,a,x1,b,y1,c,z13 is the following set:(13)Aα=x,y,z∈ℝ3|x−x1ah−α2+y−y1bh−α2+z−z1ch−α2≤1.Definition 5.
A 3-dimensional fuzzy numberA defined on ℝ3 is called convex fuzzy number if for all α∈0,1, the α-cuts(14)Aα=x,y,z∈ℝ3|μAx,y,z≥α,are convex subsets in ℝ3.Definition 6 (see [26]).
LetA and B be two continuous convex fuzzy numbers defined on ℝ3.(1)
Parametric additionA+pB:(15)A+pBα=f1αs+g1αs,f2αs,t+g2αs,t,f3αs,t+g3αs,t∈ℝ3,0≤s≤2π,−π2≤t≤π2.(2)
Parametric subtractionA−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 multiplicationA⋅pB:(17)A⋅pBα=f1αs⋅g1αs,f2αs,t⋅g2αs,t,f3αs,t⋅g3αs,t∈ℝ3|0≤s≤2π,−π2≤t≤π2.(4)
Parametric divisionA/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 3.
LetA=h1,a1,x1,b1,y1,c1,z13 and B=h2,a2,x2,b2,y2,c2,z23 be two generalized 3-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(19)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(20)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(21)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(22)xαs=x1+a1h1−αcossx2−a2h2−αcoss,yαs.t=y1+b1h1−αsinscosty2−b2h2−αsinscost,zαs.t=z1+c1h1−αsinssintz2−c2h2−αsinssint.
Furthermore, we have(23)A∗pB0=limα⟶0+A∗pBα,∗=+,−,⋅,/,A∗pBh1=limα⟶h1−A∗pBα,∗=+,−,⋅,/.
Ifh1<α≤h2, by Zadeh’s max-min principle operations, we get(24)A∗pBα=∅,∗=+,−,⋅,/.Proof.
SinceA and B are continuous convex fuzzy sets defined on ℝ3, by Theorem 3.3 in [26], there exists f1αs,g1αsfiαs,t,giαs,t,i=2,3 such that(25)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.
SinceA=h1,a1,x1,b1,y1,c1,z13 and B=h2,a2,x2,b2,y2,c2,z23, we have(26)f1αs=x1+a1h1−αcoss,f2αs,t=y1+b1h1−αsinscost,f3αs,t=z1+c1h1−αsinssint,if0≤α≤h1,g1αs=x2+a2h2−αcoss,g2αs,t=y2+b2h2−αsinscost,g3αs,t=z2+c2h2−αsinssint,if0≤α≤h2.(1)
If0<α<h1, since(27)f1αs+g1αs=x1+x2+a1h1−α+a2h2−αcoss,f2αs,t+g2αs,t=y1+y2+b1h1−α+b2h2−αsinscost,f3αs,t+g3αs,t=z1+z2+c1h1−α+c2h2−αsinssint,we have(28)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.Furthermore, we have(29)A+pB0=x,y,z∈ℝ3|x−x1−x2a1h1+a2h22+y−y1−y2b1h1+b2h22+z−z1−z2c1h1+c2h22=1,A+pBh1=x,y∈ℝ2|x−x1−x2a2h2−h12+y−y1−y2b2h2−h12+z−z1−z2c2h2−h12=1,A+pBα=∅,h1<α≤h2.(2)
If0≤s≤π,−π/2≤t≤π/2 and 0<α<h1,(30)f1αs−g1αs+π=x1−x2+a1h1−α+a2h2−αcoss,f2αs,t−g2αs+π,t=y1−y2+b1h1−α+b2h2−αsinscost,f3αs,t−g3αs+π,t=z1−z2+c1h1−α+c2h2−αsinssint.In the case ofπ≤s≤2π,−π/2≤t≤π/2, we have(31)f1αs−g1αs−π=f1αs−g1αs+π,f2αs,t−g2αs−π,t=f2αs,t−g2αs+π,t,f3αs,t−g3αs−π,t=f3αs,t−g3αs+π,t.Thus,(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.Furthermore, we have(33)A−pB0=x,y,z∈ℝ3|x−x1+x2a1h1+a2h22+y−y1+y2b1h1+b2h22+z−z1+z2c1h1+c2h22=1,A−pBh1=x,y,z∈ℝ3|x−x1+x2a2h2−h12+y−y1+y2b2h2−h12+z−z1+z2c2h2−h12=1,A−pBα=∅,h1<α≤h2.(3)
LetA⋅pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2. From the definitions of f1αs,g1αs,fiαs,t, and giαs,ti=2,3, we have(34)xαs=x1x2+x1a2h2−α+x2a1h1−αcoss+a1a2h1−αh2−αcos2s,0<α<h1,yαs,t=y1y2+y1b2h2−α+y2b1h1−αsinscost+b1b2h1−αh2−αsin2scos2t,0<α<h1.zαs,t=z1z2+z1c2h2−α+z2c1h1−αsinssint+c1c2h1−αh2−αsin2ssin2t,0<α<h1.Furthermore, we have(35)x0s=x1x2+x1a2h2+x2a1h1coss+a1a2h1h2cos2s,xh1s=x1x2+x1a2h2−h1coss,y0s,t=y1y2+y1b2h2+y2b1h1sinscost+b1b2h1h2sin2scos2t,yh1s,t=y1y2+y1b2h2−h1sinscost,z0s,t=z1z2+z1c2h2+z2c1h1sinssint+c1c2h1h2sin2ssin2t,zh1s,t=z1z2+z1c2h2−h1sinssint,A⋅pBα=∅,h1<α≤h2.(4)
LetA/pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2. Similarly, if 0<α<h1,(36)xαs=x1+a1h1−αcossx2−a2h2−αcoss,yαs.t=y1+b1h1−αsinscosty2−b2h2−αsinscost,zαs.t=z1+c1h1−αsinssintz2−c2h2−αsinssint.
Furthermore, we have(37)x0s=x1+a1h1cossx2−a2h2coss,xh1s=x1x2−a2h2−h1coss,y0s,t=y1+b1h1sinscosty2−b2h2sinscost,yh1s,t=y1y2−b2h2−h1sinscost,z0s,t=z1+c1h1sinssintz2−c2h2sinssint,zh1s,t=z1z2−c2h2−h1sinssint,A/pBα=∅,h1<α≤h2.
The proof is complete.Example 1.
LetA=1/2,6,3,8,5,4,73 and B=2/3,4,2,5,3,6,43. Then, by Theorem 3, we have the following.(1)
For0<α<1/2, the α-set of A+pB is(38)x,y,z∈ℝ3|3x−1517−30α2+3y−2422−39α2+z−116−10α2=1.(2)
For0<α<1/2, the α-set of A−pB is(39)x,y,z∈ℝ3|3x−317−30α2+3y−622−39α2+z−36−10α2=1.(3)
For0<α<1/2, A⋅pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2, where(40)xαs=6+14−24αcoss+41−2α2−3αcos2s,yαs,t=15+863−49αsinscost+201−2α23−αsin2scos2t,zαs,t=28+36−58αsinssint+41−2α2−3αsin2ssin2t.(4)
For0<α<1/2, A/pBα=xαs,yαs,t,zαs,t|0≤s≤2π,−π/2≤t≤π/2, where(41)xαs=9+91−2αcoss6−42−3αcoss,yαs,t=15+121−2αsinscost9−152−3αsinscost,zαs,t=7+21−2αsinssint4−22−3αsinssint.


## 4. Conclusion

When there are several uncertain elements, definitions and theories of the algebraic operator on fuzzy sets defined in 3-dimensional space are required and they should be natural extensions of the one-dimensional and 2-dimensional concepts. In Definition6, we defined the algebraic operator as an extension of the 2-dimensional case in 3-dimensional space. In Theorem 3, we calculated the algebraic operators on the generalized triangular fuzzy set. Using this theorem, we obtained the results for the 3-dimensional generalized triangular fuzzy sets A and B in Example 1, and the graphs were presented.Graphs for results obtained in the example were drawn and the Mathematica commands were presented for each. Each graph of A and B was cut with a plane passing through its central axis, and A2 and B2 show the result of the cut (Figures1–4).(: pre-run :)PP = 100; IS = 360; legend = Placed[BarLegend[Automatic, LegendMarkerSize -> 300], Right];Figure 1
A.Figure 2
A2.Figure 3
B.Figure 4
B2.Among the four results of the algebraic operator, six graphs from 0 cut to 1/2 cut for plus were presented (Figures5 and 6). Also, with the commands introduced in the paper, researchers will be able to obtain cuts or information of their need.(∗ (A + B) 1∼6 ∗)p = 17; q = 22; r = 6;g[a_]: = ContourPlot3D[((3x − 15)/(p − 30a))^2 + ((3y − 24)/(q − 39a))^2 + ((z − 1)/(r − 10a))^2 = = 1,{x, −4, 14}, {y, −1, 17}, {z, 2, 20}, ContourStyle -> Directive[RGBColor[1 − a, 1 − a, 1 − a], Opacity[0.1]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5}, BoundaryStyle -> None, ImageSize -> IS];tg = Column[Table[g[a] ,{a, 0, 0.5, 0.1}]];Show[tg](∗ (A + B)/2 1∼6 ∗)p = 17; q = 22; r = 6;g[a_]: = ContourPlot3D[((3x − 15)/(p − 30a))^2 + ((3y − 24)/(q − 39a)) ^ 2 + ((z − 11)/(r − 10a))^2 = = 1,{x,−4, 14},{y, −1, 17},{z, 2, 11}, ContourStyle -> Directive[RGBColor[1 − ,1 − a, 1 − a], Opacity[0.1]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5/2}, BoundaryStyle -> None, ImageSize -> IS];tg = Column[Table[g[a], {a, 0, 0.5, 0.05}]];Show[tg].(∗A + B∗)p = 17; q = 22; r = 6;g[a_]: = ContourPlot3D[((3x − 15)/(p − 30a))^2 + ((3y − 24)/(q − 9a)) ^ 2 + ((z − 11)/(r − 10a))^2 = = 1, {x, −4, 14}, {y, −1, 17}, {z, 2, 20}, ContourStyle -> Directive[RGBColor[1 − a, 1 − a, 1 − a], Opacity[0.2]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5}, BoundaryStyle -> None, ImageSize -> IS, ViewPoint -> {p, 2q, 1.2r}];tg = Table[g[a], {a, 0, 0.5, 0.05}];Show[tg](∗ (A + B)/2 ∗)p = 17; q = 22; r = 6;g[a_]: = ContourPlot3D[((3x − 15)/(p − 30a))^2 + ((3y − 24)/(q − 39a)) ^ 2 + ((z − 11)/(r − 10a))^2 = = 1,{x, −4, 14},{y, −1, 17},{z, 2, 11}, ContourStyle -> Directive[RGBColor[1 − a, 1 − a, 1 − a], Opacity[0.2]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5/2}, BoundaryStyle -> None, ImageSize -> IS, ViewPoint -> {p, 2q, 1.2r}];tg = Table[g[a],{a, 0, 0.5, 0.05}];Show[tg](∗A − B∗)p = 17; q = 22; r = 6;g1[a_]: = ContourPlot3D[((3x − 3)/(p − 30a))^2 + ((3y − 6)/(q − 39a)) ^ 2 + ((z − 3)/(r − 10a))^2 = = 1, {x, 1 − (p − 30a)/3, 1 + (p − 30a)/3}, {y, 2 − (q − 39a)/3, 2 + (q − 39a)/3}, {z, 3 − (r − 10a), 3 + (r − 10a)}, ContourStyle -> Directive[RGBColor [1 − a, 1 − a, 1 − a], Opacity[0.1]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5},BoundaryStyle -> None, ImageSize -> IS];tg = Table[g1[a], {a, 0, 0.5, 0.1}];Show[tg](∗ (A − B)/2 ∗)p = 17; q = 22; r = 6;g1[a_]: = ContourPlot3D[((3x − 3)/(p − 30a))^2 + ((3y − 6)/(q − 39a)) ^ 2 + ((z − 3)/(r − 10a))^2 = = 1, {x, 1 − (p − 30a)/3, 1 + (p − 30a)/3}, {y, 2 − (q − 39a)/3, 2 + (q − 9a)/3}, {z, 3 − (r − 10a), 3}, ContourStyle -> Directive[RGBColor[1 − a, 1 − a, 1 − a], Opacity[0.1]], Mesh -> None, PlotPoints -> PP, BoxRatios -> {1, 1, 0.5/2}, BoundaryStyle -> None, ImageSize -> IS];tg = Table[g1[a], {a, 0, 0.5, 0.1}];Show[tg].Figure 5
A+B,0−cut∼1/2−cut.Figure 6
A+B/2,0−cut∼1/2−cut.In general, it is not easy to express the results defined in 3D in graphs. Mathematica commands in this paper will help facilitate better visual understanding. Algebraic operators extended in 3D were defined, and they were calculated for a general trigonometric fuzzy set. Also, the resultant graphs can be specifically applied to fuzzy logic and fuzzy control (Figures7–10). In sum, this paper expanded the results of two-dimensional cases and will provide the theoretical base for further development of 4-dimensional cases.Figure 7
A+B,A+B/2.Figure 8
A−B,A−B/2.Figure 9
AB.Figure 10
A/B.



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