The document discusses mathematical operations and intervals involving real numbers. Some key points: - Various mathematical operations like addition, subtraction, multiplication and division are described when the operands are within given intervals. For example, if a is in [a,b] and c is in [c,d], then a+c is in [a+c, b+d]. - Properties of intervals are defined, such as the inverse of a number in [a,b] being in [1/b, 1/a]. - Material properties like stiffness, density and tensile strength for 5 materials fall within given numerical ranges. - A method is described to calculate the similarity between data points using their

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3. DENG •
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4. •
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5. •
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⊗∈[a,∞] •
⊗∈(−∞,a) •
⊗∈[a,a] •
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6. ⊗1 ∈ [a, b], a〈b;
⊗ ∈ [c, d ], c〈d;
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⊗ + ⊗2∈[a + c,b + d ];
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− ⊗ = [−b,−a];
⊗1 − ⊗2 = ⊗1 + (− ⊗2 ) ∈ [a − d, b − 7
c];

7. • ab〉 o ⊗-1 ∈ [1/b,1/a];
• ⊗1 ⋅ ⊗2 ∈ [min{ac, ad , bc, bd}, max{ac,
ad , bc, bd}];
• K ∈ℜ + K ⋅⊗∈[Ka, Kb]
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8. •
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9. ξ∈ (o,1) •
ɣoi ɣ (Xo , Xi )
k ɣ (xo (k), xi (k))
ɣoi (k)
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10. •
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3 •
4 •
5 •
6 •
7 •
8 •
11

11. (g/cm3) (Mpa) (GPA)
[530, 580] [10, 20] [6, 9] [180,200] [25,35]
[380, 420] [10, 15] [1.2,4.2] [80,100] [5,15]
[330, 380] [1, 5] [1, 4] [3400,3600] [67,78]
[470, 530] [1, 2] [0.5, 3] [3900,4100] [345,355]
[220, 280] [9, 12] [0.5,2] [3500,3700] [115,125]
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12. •
• Wl ∊ [0.10,0.10],
• w2 ∊[0.20,0.20],
• w3 ∊[0.20,0.20],
• w4 ∊[0.30, 0.30],
• w5 ∊[0.40,0.40]
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13. •
Stiffness Tensile
strength
Density Elongation
at break
Max temp •
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||Ai|| 355 4100 9 20 580
||x||=max(max(|a1-|,|a1+|), max(|a2-|,|a2+|),…, max(|a3-|,|a3+|) 14

14. •
Stiffness Tensile Density Elongation Max temp
strength at break
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Wi [0.1,0.1] [0.2,0.2] [0.2,0.2] [0.3,0.3] 8 [0.4,0.4]
X3,2
X3,2 [ =[0.8293,0.8781]
||x||=max(max(180,200),) max(80,100,
max(3400,3600),max(3900,4100), max(3500,3700) 15

15. •
G1 G2 G3 G4 G5
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A1
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A2
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A3
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A4 •
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A5 •
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Wi [0.1,0.1] [0.2,0.2] [0.2,0.2] [0.3,0.3] [0.4,0.4]
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X3,2
[0.8293*0.2,0.8781*0.2 =[0.1658,0.1756]
X3,2
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16. •
G1 G2 G3 G4 G5
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A1
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A2
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A3
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A4 •
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A5 •
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U0 [0.0972,0.1] [0. 9 ,0.2] [0.133,0.2] [0.15,0.225] [0.365,0.4]
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X3,2
=(ro2 - -0.1658)+(0.2-0.1756) 0.0488
∆3,2 r3,2)= 18

18. •
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ɣ(xo(k),x1(k)) 0.5383 0.3614 1.0000 1.000 1.0000
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ɣ(xo(k),x2(k)) 0.5233 0.3554 0.4918 0.7372 0.4959 •
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ɣ(xo(k),x3(k)) 0.5734 0.8116 0.4853 0.3688 0.4327
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ɣ(xo(k),x4(k)) 1.0000 1.0000 0.4514 0.3418 0.7348
ɣ(xo(k),x5(k)) (0.6187) 0.8436 0.4309, 0.6091 0.3333
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 ɣ(xo(2),x3(2))= (0+0.5*(0.4207))/(0.0488+(0.5)*(0.4207))=0.8116

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ɣ(xo,x1)=1/5 *(0.5383, 0.3614,1.0000,1.0000,1.0000)=0.7799
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ɣ(xo,x1)=1/5 *(0.5233, 0.3554, 0.4918, 0.7372, 0.4959)=
0.5187 6
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ɣ(xo,x1)=1/5 *(0.5734, 0.8117, 0.4853, 0.3688, 0.4327)= •
0.5344 8
ɣ(xo,x1)=1/5 *(1.0000,1.0000, 0.4514, 0.3418, 0.7348)=
0.7056
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ɣ(xo,x1)=1/5 *(0.6187, 0.8436, 0.4309, 0.6091,

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