Diplomarbeit: EAP und OAP im Fuzzy-Umfeld - Institut für Informatik

0 1
0 1
[0, 1]
x ∈ [0, 1]
1 − x
1
Wahrheitswert
0
Aussage x
1
Wahrheitswert
0
Aussage x
M
M = {m1, . . . , mn}

M X
M
⎧
⎨1,
x ∈ M
χM(x) =
⎩0,
χM(x)
M x ∈ X
µM(x) ∈ [0, 1] x ∈ X
M = {(x, µM(x)) | x ∈ X}.
M1 M2
M1 = M2 ⇐⇒ µM1 (x) = µM2 (x) ∀x ∈ X.
M X
M supp(M) = {x ∈ X | µM(x) > 0}
α α M acutα(M) = {x ∈ X | µM(x) ≥ α}
M core(M) = {x ∈ X | µM(x) = 1}
M bnd(M) = {x ∈ X | 0 < µM(x) < 1}

M hgt(M) = sup µM(x)
x∈X
M
M
supp(M) = core(M)
M
x0
supp(M) = core(M) = {x0},
M M1 M2 X
µM1∪M2 (x) = max(µM1 (x), µM2 (x))
µM1∩M2 (x) = min(µM1 (x), µM2 (x))
M M µ M (x) = 1 − µM(x)
M1 M2 µM1 (x) ≤ µM2 (x) ∀x ∈ X
M = {(m1, µM(m1)), (m2, µM(m2)), . . . , (mn, µM(mn))} :=
n µM(mi)
,
M =
m∈M
µM(m)
m .
i=1
mi

1
0
µ
Kern
a b c d
Übergang
Träger
µT T
T = [a, b, c, d] a ≤ b ≤ c ≤ d
⎧
1, x ∈ [b, c]
⎪⎨ x−a
b−a , x ∈ [a, b)
µT (x) =
d−x
d−c , x ∈ (c, d]
⎪⎩ 0,
[b, c] b = c
Π
1
1
hgt(M) = 1
M ⊆ X
x

1
0
µ
a b c d
1
0
µ
a b=c d
1
0
µ
a b
Π
T = [A, B, a, b] A ≤ B a, b ≥ 0
⎧
1,
⎪⎨ 1 −
µT (x) =
⎪⎩
x ∈ [A, B]
A−x
a ,
1 −
x ∈ [A − a, A)
x−B
b ,
0,
x ∈ (B, B + b]
A B a
b
1
0
µ
A-a a A B b B+b
x D y
x, y ∈ D
x

ζ(x, x) = 1
ζ(x, y) = ζ(y, x)
ζ(x, z) ≥ max min{ζ(x, y), ζ(y, z)}
y∈D
ζ(x, z) ≥ max{ζ(x,
y) · ζ(y, z)}
y∈D
ζ(x, y) ∈ [0, 1]
x y z D
schlecht
normal
gut
schlecht
1
0.6
0.2
normal gut
0.6 0.2
1
0.3
0.3
1

ζ(x, x) = 1
ζ(x, y) = ζ(y, x)
ζ(x, z) ≥ max min{ζ(x, y), ζ(y, z)}
y∈D
ζ(x, z) ≥ max{ζ(x,
y) · ζ(y, z)}
y∈D
ζ(x, y) ∈ [0, 1]
x y z D
schlecht
normal
gut
schlecht
1
0.6
0.2
normal gut
0.6 0.2
1
0.3
0.3
1

T = [a, b, c, d] a ≤ b ≤ c ≤ d
M mi
µ(mi) M = {(m1, µ(m1)), . . . , (mn, µ(mn))}
m1, . . . , mn µ(mi) ∈ (0, 1] i = 1, . . . , n
ζ(·, ·)

α β S1(α, β)
α β P aT N3sup(α, β)
α β P aT N2rel(α, β)
α β InSN2rel(α, β)
α β P aSN3inf (α, β)
α β NK3(α, β)
α β F GEQ(α, β)
α β F GT (α, β)
α β NF GEQ(α, β)
α β NF GT (α, β)
α β NK3(β, α)
α β F GEQ(β, α)
α β NF GEQ(β, α)
α β F GT (β, α)
α β NF GT (β, α)

α β S1(α, β)
α β P aT N3sup(α, β)
α β P aT N2rel(α, β)
α β InSN2rel(α, β)
α β P aSN3inf (α, β)
α β NK3(α, β)
α β F GEQ(α, β)
α β F GT (α, β)
α β NF GEQ(α, β)
α β NF GT (α, β)
α β NK3(β, α)
α β F GEQ(β, α)
α β NF GEQ(β, α)
α β F GT (β, α)
α β NF GT (β, α)

α < β α β 1.0
α β α β 1.0
α >= β α β
γ α β γ [α, α, β, β]
α β γ R α β α
β ∈ R α [α, α, α, α] β [β, β, β, β]

Q
Q
Q
Q = P P
Q
Q = P1 op · · · op Pn
P1, . . . , Pn op
∧
∨
Q = P1 op P2 op · · · op Pn Q ′ Q
Q ′ = Ps1 op Ps2 op · · · op Psr {s1, . . . , sr} ⊆ {1, . . . , n} Q ′
{s1, . . . , sr} ⊂ {1, . . . , n}

Q
Q
Q
Q = P P
Q
Q = P1 op · · · op Pn
P1, . . . , Pn op
∧
∨
Q = P1 op P2 op · · · op Pn Q ′ Q
Q ′ = Ps1 op Ps2 op · · · op Psr {s1, . . . , sr} ⊆ {1, . . . , n} Q ′
{s1, . . . , sr} ⊂ {1, . . . , n}

Q
Q
Q
Q = P P
Q
Q = P1 op · · · op Pn
P1, . . . , Pn op
∧
∨
Q = P1 op P2 op · · · op Pn Q ′ Q
Q ′ = Ps1 op Ps2 op · · · op Psr {s1, . . . , sr} ⊆ {1, . . . , n} Q ′
{s1, . . . , sr} ⊂ {1, . . . , n}

Q
Q
Q
Q = P P
Q
Q = P1 op · · · op Pn
P1, . . . , Pn op
∧
∨
Q = P1 op P2 op · · · op Pn Q ′ Q
Q ′ = Ps1 op Ps2 op · · · op Psr {s1, . . . , sr} ⊆ {1, . . . , n} Q ′
{s1, . . . , sr} ⊂ {1, . . . , n}

Q
Q
Q
Q = P P
Q
Q = P1 op · · · op Pn
P1, . . . , Pn op
∧
∨
Q = P1 op P2 op · · · op Pn Q ′ Q
Q ′ = Ps1 op Ps2 op · · · op Psr {s1, . . . , sr} ⊆ {1, . . . , n} Q ′
{s1, . . . , sr} ⊂ {1, . . . , n}

1
0
µ
22 24 26 28 30 32 Alter
α

1
0
µ
20 22 28 30 Alter

1
0
µ
20 22 28 30 Alter

+
ν

P
∀x ∈ dom(A) µ mehr−oder−weniger(P )(x) ≥ µP (x) A
P dom(A)
A
supp(P ) ⊂ supp(
(P )) {x | µP (x) > 0} ⊂ {x | µ mehr−oder−weniger(P )(x) > 0}
m
f µf
[A, B, a, b, ε] f(x) = ε x ≤ A − a x ≥ B + b
f(x) = 1 A ≤ x ≤ B f(x) = f ′ (x) A − a ≤ x ≤ A f(x) = f ′′ (x)
B ≤ x ≤ B + b f ′ f ′′ f ′ x < A
f ′′ x > B f ′ (A) = f ′′ (B) = 1 f ′ (A − a) = f ′′ (B + b) = ε
f ϕ ϕ(x) = 1
A ≤ x ≤ B ϕ(x) = f ′ (x) x ≤ A ϕ(x) = f ′′ (x) x ≥ B
m
∀x ∈ U, g(x) = m(f(x)) = min(1, max(0, α · ϕ(x) + β)),
U α, β ∈ R
g [A ′ , B ′ , a ′ , b ′ , ε ′ ]
ν
α = ν ∈ [0.5, 1)
β = 1 − ν ∀x ∈ U g(x) = max(0, ν · ϕ(x) + 1 − ν) g
[A, B, a ′ , b ′ , ε] = [A, B, a/ν, b/ν, ε] a ′ = a/ν a ′

1
1-ѵ
0
µ
ѵ-rather(T)
A-a/ѵ
A-a A B
supp(T)
B+b
supp(ѵ-rather(T))
T
B+b/ѵ
ν
a ′ = a + θ · a b ′ = b + θ · b θ
θ = (1 − ν)/ν ∈ (0, 1]
core(f) = core(g)
Q
P Q = P
ν Q
Q1 = P1 = ν(P )
n
Qn = ν(Qn−1) = ν(ν(. . . ν(P ) . . .)).
U

1
0
µ
F P
Q i
Q F P
FP
FP
x i µ(x) = min(µQi (x), 1 − µ FP (x))
FP
= ∅
Qi
Qi FP core(Qi) ⊂ core(FP )
Qi FP
∀x min(µQi (x), 1 − µ FP (x)) = 0
Qi FP
FP
FP
a ′ = a/ν = a b ′ = b/ν = b a = b = 0 ν

θ ∈ (0, 1]
Qi
Q ′ i Qi
∩ FP
Q ′
i
∅ core(Qi) ⊆ core(FP )
Qi+1 ν(Qi, θ)
Q ′ i Qi ∩ FP
Q ′ i
= ∅
ν FP
Q = P1 ∧ · · · ∧ Pn Q ′ = Ps1 ∧
· · ·∧Psr {s1, . . . , sr} ⊂ {1, . . . , n} Q
≤ ∧ =
∧
≤ ∧ = ∧

Q5
(20, m, 180)
Q2
(19, m, 180)
Q6
(19, *, 180)
Q1
(18, m, 180)
Q3
(18, *, 180)
Q7
(19, m, 179)
Q8
(18, *, 179)
Q4
(18, m, 179)
Q9
(18, m, 178)
Q5 Q2 Q5 =
(20, m, 180) Q2
⇔
Q = ∅
∧
∪

Q
Q1
|∑|=0
(P , P , P )
1 2 3
Q |∑|=25
2
Q3
|∑|=0 Q |∑|=0
4
(P , P )
(P , P )
(P , P )
2 3
1 3
1 2
Q5
|∑|=25 Q6
|∑|=40 Q7
|∑|=5
(P )
(P )
(P )
3 2 1
Q = P1∧P2∧P3
P2

Q
Q
Q = ∅ Qi Q
P1 ∧P2, P1 ∧P3
O(N)
∅
⇔
Q = ∅
∅
∅
∈
− ∪
−
− ∪
k k < n n
Q = P1 ∧ · · · ∧ Pn

T ↑ P
ν
ν
T ↑ (P ) P
T ↑ ∀x ∈
dom(A), µ T ↑ (P )(x) ≥ µP (x) A P
dom(A) A
T ↑ P supp(P ) ⊂ supp(T ↑ (P ))
T ↑
P core(P ) = core(T ↑ (P ))
T ↓
ν
R
U
µR(x, x) = 1
µR(x, y) = µR(y, x)
x, y ∈ U µR(x, y) x y
x y
x/y 1
x − y 0

x y µCl
µCl(x, y) = µM(x/y)
M
µM(1) = 1
µM(x) = 0 x ≤ 0
µM(x) = µM(1/x)
µCl(x, y) = µCl(y, x)
M supp(M) = [1 − ε, 1/(1 − ε)]
ε ∈ R M =
[1, 1, ε, ε/(1 − ε)] µM(x/y) = 1
x = y µM(x/y) = 0 x y M = [1, 1, 0, 0]
Cl[M] M
1
0
µ
ε 1 ε/(1-ε)
M
Ne x y
R

x + y y x y x
y
µNe(x, y) = µCl(x + y, y) = µM(1 + x/y).
supp(M) M
V = [( √ 5 − 1)/2, ( √ 5 + 1)/2] ≈ [0.61, 1.61]
Cl
Ne
µCl(x, y) ≤ 1 − max(µNe(x, y), µNe(y, x)).
M ε
[0, (3 − √ 5)/2] ≈ [0, 0.38] M
V
V
Cl
P P ′
∀x ∈ U µP
′(x) = sup(min(µP
(y), µ CL[M](y, x)))
y∈U
= sup(min(µP
(y), µM(y/x))).
y∈U
M M = 1/M
= sup(min(µP
(y), µM(x/y))).
y∈U
⊗
µP ′(x) = µP ⊗M(u).

P ′ P ′ = T ↑ (P ) = P ⊗ M
P ′
CL[M] P ′
P P
P = [A, B, a, b]
M = [1, 1, ε, ε/(1 − ε)]
P ′ = [A, B, a + |A| · ε, b + |B| · ε/(1 − ε)] = [A, B, a + ∆l(ε), b + ∆r(ε)] = [A, B, a ′ , b ′ ].
P
1
0
µ
A-a’
P’
A-a A B B+b
a+Δ (ε)
l
b+Δ (ε)
r
P
B+b’
T ↑ P
∆l(ε) = |A| · ε ∆r(ε) = |B| · ε/(1 − ε)
∆r(ε) > ∆l(ε) A, B ≥ 0
T ↑max (P ) P εmax = (3 − √ 5)/2
P ′ max = T ↑max (P ) = [A, B, a +
∆l(εmax), b + ∆r(εmax)]
Q = Q0 P
T ↑ n
Qn = T ↑(n) (P ) = P ⊗ M n M n
U

supp(M i ) = [(1 − ε) i , 1/(1 − ε) i ] i = 2 M 2 = M ⊗ M =
[1, 1, 2 · ε − ε 2 , (2 · ε − ε 2 )/(1 − ε) 2 ] supp(M 2 ) = [(1 − ε) 2 , 1/(1 − ε) 2 ]
M i
V
supp(M i ) ⊂ V
ε ∈ [0, (3 − √ 5)/2]
Q Qi
i
ε
Q0
Q i = ∅ supp(M i+1 ) ⊆ V
Qi P ⊗ M i
Q i
ν
P = [A, B, a, b] core(P ) A
B A B
P1 = [A1, B1, a, b] P2 = [A2, B2, a, b]
a b a = b
0 ≤ A1 < A2 0 ≤ B1 < B2
P ′ 1 = T ↑ (P1) = [A1, B1, a ′ 1 , b′ 1 ] P ′ 2 = T ↑ (P2) =
[A2, B2, a ′ 2 , b′ 2 ] a′ 1 = a+A1·ε, b ′ 1 = b+B1·ε/(1−ε), a ′ 2 = a+A2·ε, b ′ 2
a ′ 1 < a′ 2 b ′ 1 < b′ 2 a ′ 1 < b′ 1 a ′ 2 < b′ 2
= b+B2·ε/(1−ε)
ν

1
0
µ
P’
1
P
1
A -a’ A -a A B B +b B +b’ A -a’ A -a A B B +b B +b’
1 1 1
1 1 1
1 1 2 2 2 2 2 2
2 2
P = [A, B, 0, 0] P ′ = [A, B, |A| · ε, |B| · ε/(1 − ε)]
P ′ = ν(P ) = P
P = [−5, −5, 0, 0] ε = 0.1
T ↑ (P ) = [−5, −5, | − 5| · 0.1, | − 5| · 0.1/0.9] ≈ [−5, −5, 0.5, 0.56]
∆l ∆r
A, B < 0 T ↑ (P )
P = [A, B, a, b] P ′ = [−B, −A, b, a]
T ↑ (P ′ ) = P ′ ⊗ M = [−B, −A, b + (−B) · ε, a + (−A) · ε/(1 − ε)]
[−B, −A, b+(−B)·ε, a+(−A)·ε/(1−ε)]
[A, B, a + (−A) · ε/(1 − ε), b + (−B) · ε]
A < 0 B >= 0
P’
2
P
2
U

ε ε ∈ [0, (3 − √ 5)/2]
ε n
P = [A, B, a, b]
n P n = T ↑(n) (P ) = [A, B, a + |A| · n · ε, b + |B| · n ·
ε/(1 − ε)] ε εmax = (3 − √ 5)/2) ≈ 0.38
ε = εmax/n
ν
FP
P = [20, 22, 0, 3]
FP = [18, 30, 0, 5]
µ(x) = min(µP (x), 1 − µFP (x)) ∀x ∈ dom(P ).
M i V
T ↑ (P ) FP
core(T ↑ (P )) ⊂ core(FP ).
ν
Pi Q = P1 ∧ · · · ∧ Pn

k P1 ∧ · · · ∧ Pk
εmax = (3 − √ 5)/2
ε = εmax/n
Q i
Qi
Q0
Q i = ∅
Pj
Qi ∧ · · · ∧
Q i
[1, 1, ε, ε/(1 − ε)]
P ′ = P ⊗ M h
P ′
ε
P ′
V n
P ′ = T ↑ (P 1) ∧ P2 ∧ · · · ∧ Pn
P ′′ = T ↑ (P 1) ∧ T ↑ (P2) ∧ · · · ∧ T ↑ (Pn) P ′
P2

Pi Q = P1 ∧ · · · ∧
Pn
T ↑ (·)
Q1 Q2 n P1 Pn
Q1 ≺ Q2 ⇐⇒
n
i=1
anzahl(T ↑
i in Q1) <
n
i=1
anzahl(T ↑
i
in Q2).
T ↑
1 T ↑ n
P ′
i Pi
∆i(Pi, T ↑
i (Pi)) = laenge(T ↑
i (Pi)) / laenge(Pi)
laenge([A, B, a, b]) = B − A + a + b
Pi = [A, A, 0, 0]
laenge(Pi) = 0
≺
Q = P1 ∧ P2
V
ε = εmax/n
2 · n + 1
n = 3 Q = P1∧P2
T ↑(3) (Q) = T ↑(3)
1
Q = P1 ∧ · · · ∧ Pn
T ↑(max) (Q) = T ↑(max)
1
(P1) ∧ T ↑(3)
2
(P1) ∧ · · · ∧ T ↑(max)
(Pn).
n
(P2)
ε

↑(3)
T (P ) 1 1
Ʌ P 2
↑(2)
T (P ) 1 1
Ʌ P 2
↑(2)
T (P ) 1 1
↑(1)
T (P ) 1 1
↑(1)
Ʌ T (P ) 2 2
Q
P Ʌ P
1 2
↑(1)
Ʌ P P 2 1 Ʌ T (P ) 2 2
↑(1)
T (P ) 1 1
↑(1)
↑(2)
Ʌ T (P )
P1 Ʌ T (P )
2 2
2 2
↑(1)
T (P ) 1 1
↑(2)
↑(3)
Ʌ T (P )
P1 Ʌ T (P )
2 2
2 2
P1 Pn ∆i
Pi εi
∆1(P1, T ↑
1 (P1)) = · · · = ∆n(Pn, T ↑ n(Pn)).
εi i = 1, . . . , n Pj j ∈ {0, . . . , n}
εj εi, 0 < i ≤ n, i = j
Pj
Q
Q

↑(3)
T (P ) 1 1
Ʌ P 2
↑(3)
T (P ) 1 1
↑(2)
T (P ) 1 1
Ʌ P 2
↑(2)
T (P ) 1 1
↑(1)
Ʌ T (P ) 2 2
↑(1)
T (P ) 1 1
↑(3)
T (P ) 1 1
↑(1)
Ʌ T (P ) 2 2
Q
P Ʌ P
1 2
↑(1)
Ʌ P P 2 1 Ʌ T (P ) 2 2
↑(1)
T (P ) 1 1
↑(2)
T (P ) 1 1
↑(2)
Ʌ T (P ) 2 2
↑(3)
T (P ) 1 1
↑(1)
↑(2)
Ʌ T (P )
P1 Ʌ T (P )
2 2
2 2
↑(2)
Ʌ T (P ) 2 2
↑(1)
T (P ) 1 1
↑(2)
T (P ) 1 1
↑(3)
Ʌ T (P ) 2 2
↑(2)
↑(3)
Ʌ T (P )
P1 Ʌ T (P )
2 2
2 2
↑(1)
T (P ) 1 1
↑(3)
Ʌ T (P ) 2 2
↑(3)
Ʌ T (P ) 2 2
k
mfs(Q) Q
Q = P1 ∧ P2 ∧ P3 mfs(Q) = {P1 ∧ P2} T ↑
3 (P3) Q ′ = P1 ∧
P2 ∧ T ↑
1 (P3) mfs(Q)
Q ′′ = T ↑
1 (P1) ∧ P2 ∧ P3 Q ′ {T ↑
1 (P1) ∧ P2}
Q
Q ′ mfs(Q)

P1 ∧ · · · ∧ Pk
Q = ∅
/∈
∅
= ∅
Qmax
Qmax
Q
Q ⊂ ⊂ Qi Qmax
i < max
Qi

x y x/y
1
x − y 0
E[Z]
Z
x y
µ E[Z](x, y) = µZ(x − y).
Z
µZ
µZ(0) = 1
µZ(x) = µZ(−x)
Z supp(Z) = [−δ, δ] δ ∈ R, δ ≥ 0
µE(x, y) = µZ(x −
y) = 0 x = y µE(x, y) = µE(y, x)
µE(x, y) < 1 x = y δ
Z δ = 0
x y µZ(x − y) = 1 x = y µZ(x − y) = 0 x = y
1
0
µ
δ 0 δ
Z
R

P P ′
∀x ∈ U µP
′(x) = sup(min(µP
(y), µ E[Z](x, y)))
y∈U
= sup(min(µP
(y), µZ(x − y)))
y∈U
= µP ⊕Z(x)
⊕
Z P ′ = P ⊕ Z
P ′ P
P Z
P = [A, B, a, b]
Z = [0, 0, δ, δ]
P ′ = T ↑ (P ) = P ◦ E[Z] = P ⊕ Z = [A, B, a + δ, b + δ] = [A, B, a ′ , b ′ ].
1
0
µ
A-a’
P’
A-a A B B+b
a+δ b+δ
P P ′ δ
P
B+b’
U

1
0
µ
P’
1
P
1
A -a’ A -a A B B +b B +b’
A -a’ A -a A B B +b B +b’
1 1 1
1 1 1 1 1 2 2 2 2 2 2 2 2
Q = P P
Q1 = P ⊕ Z
n Qn = P ⊕ n · Z
= ∅
Qn
Qi
tmax
ν
FP
= ∅
Qi
∀x ∈ U min(µQi (x), 1 − µFP (x)) = 0 i ≥ tmax
δ dom(P )
P δ
δ
δ
Q = P1 ∧ · · · ∧ Pn
Pi δi
P’
2
P
2
U

δ ∈ R,
Z = [0, 0, δ, δ]
Qi
Qi
∅ core(Qi) ⊆ core(FP )
Qi
P ⊕ i · Z
Qi
= ∅
Q
Q1 = T ↑ (Q) = T ↑
1 (P1) ∧ · · · ∧ T ↑ n(Pn)
Q
Qi = T ↑(i) (Q) = T ↑(i)
1 (P1) ∧ · · · ∧ T ↑(i)
n (Pn) i > 1

ν FP
FP
Q = P1 ∧ · · · ∧ Pn Qθ
Qθ = P1 [θ1] ∧ · · · ∧ Pn [θn] .
Pi [θi] Pi θi
x
µQθ (x) = min(θ1 · µP1 (x), . . . , θn · µPn(x)).
Qθ
Qθ θ1 θn n
i=1 θi = 1
Qθ Q
Qθ = P1 [1/n] ∧ · · · ∧ Pn [1/n] = Q
Qθ Qθ = P1 [θ1] ∧ · · · ∧ Pn−1 [θn−1] ∧
Pn [0] = P1 [θ1] ∧ · · · ∧ Pn−1 [θn−1] 0
Qθ = P1[0.9] ∧ P2[0.1]
0.1 min(0.9 ∗ 1, 0.1 ∗ 1) ≤ 0.1

Qθ
Qθ
Pi
Qθ = P1 [θ1]∧· · ·∧Pn [θn] θ1 ≥ . . . ≥ θn
µQθ (x) = (θ1 − θ2) · min(µP1 (x)) + 2 · (θ2 − θ3) · min(µP1 (x), µP2 (x)) + · · ·
µQθ (x) =
+ n · θn · min(µP1 (x), . . . , µPn(x))
n
i · (θi − θi+1) · min(µP1 (x), . . . , µPi (x)) θn+1 = 0.
i=1
µQθ
Q = P1 ∧ P2 Qθ = P1 [θ1] ∧ P2 [θ2] θ1 = θ2 =
1/2 = 0.5 Qθ = Q
µQθ (x) = (0.5 − 0.5) · min(µP1 (x)) + 2 · (0.5 − 0) · min(µP1 (x), µP2 (x))
= 2 · 0.5 · min(µP1 (x), µP2 (x))
= min(µP1 (x), µP2 (x)) = Q
θ1 > θ2 µP1 (x) ≤ µP2 (x)
µQθ (x) = (θ1 − θ2) · µP1 (x) + 2 · θ2 · µP1 (x) = (θ1 + θ2) · µP1 (x) = µP1 (x),

µP1 (x) > µP2 (x)
µQθ (x) = (θ1 − θ2) · µP1 (x) + 2 · θ2 · µP2 (x) = (1 − θ2) · µP1 (x) + 2 · θ2 · µP2 (x)
= 2 · (µP1 (x) − µP2 (x))θ1 + µP2 (x).
Qθ θ1 > θ2 P1 P2
x µP1 (x) = 0.8 µP2 (x) = 0
Q = P1 ∧ P2 x µQ(x) = min(µP1 (x), µP1 (x)) = 0
x
Q
P1
Qθ = P1 [0.6] ∧ P2 [0.4] µQθ (x) = (0.6 − 0.4) · µP1 (x) + 2 · 0.4 · µP2 (x) =
0.2 · 0.8 + 2 · 0.4 · 0 = 0.16 x 0.16
Q Q ′ θ
Qsub = P1 ∧ Pr r ≤ n Q Qsub ∈ mfs(Q)
Qθ
θi < 1/n Pi ∈ Qsub
Qθ Qθ
Q ′ = Pr+1 ∧ · · · ∧ Pn
Q = P1∧P2
P1 P2

µQ µP1 µP2
P1 P2
Q P1 P2
P1
P1
P2
P1 [θ1]
Qθ = P1 [θ1] ∧ · · · ∧ Pn [θn] θ1 ≥ . . . ≥ θn
θ1 · µP1 (x) x ∈ dom(P1) x ∈
Qθ
= ∅
Qθ
Qθ
θi
n
Qθ Qθ = P1 [θ1] ∧ · · · ∧ Pn [θn]
∆i(θi, Pi, T ↑
i (Pi)) = laenge(T ↑
i (Pi)) / (n · θi · laenge(Pi)).
n · θi · laenge(Pi)
T ↑

!

∅
Q = ( [18, 20, 0, 2] ) ∧ (∅ [0.7, 1.7, 0, 0.6] ) ∧ ( [5, 6, 0, 0] ).
P1
Q
Q
P1 P2 P3
P2
P3

1
0
µ
18
P
1
20
22
Alter
1
0
µ
0.7
P
2
1.7
2.3
Note
1
0
µ
5
P
3
6
Semester
P1 P2 P3
µQ µP1 µP2 µP3
∅
Q
Q
Q ′ ∈ mfs(Q)
MF S = ∅
p = P1 ∈ {P1, P2, P3} P2 ∧ P3
MF S ∪ {p} = {P1}
P2∧P3
= ∅ MF S =
p = P2 ∈ {P2, P3} {P3} ∪ MF S = {P3, P1} : P3 ∧ P1
= ∅ MF S
P3∧P1
p = P3 ∈ {P3} ∅ ∪ MF S : P1
MF S ∪ {p} = {P1, P3}
Q ′ = P1 ∧ P3
P1
= ∅ MF S =

P1∧P3
Q Q ′
Q ′ = P1 ∧ P3
ε P1 n
ε1 P1
ε1 = εmax/n ≈ 0.38/n.
n = 3 ε1 = 0.13
Cl[M1]
M1 = [1, 1, ε1, ε1/(1 − ε1)] = [1, 1, 0.13, 0.13/(1 − 0.13)] ≈ [1, 1, 0.13, 0.15].
P3
∆1(P1, T ↑
1 (P1)) = ∆3(P3, T ↑
3 (P3)).
∆i(Pi, T ↑
i (Pi)) = laenge(T ↑
i (Pi)) / laenge(Pi)
laenge([A, B, a, b]) = B −A+a+b P1 ∆1(P1, T ↑
1 (P1))
∆1(P1, T ↑
1 (P1)) = laenge(T ↑
1
([18, 20, 0, 2])) / laenge([18, 20, 0, 2])
= laenge([18, 20, 0, 2] ⊗ [1, 1, 0.13, 0.15]) / laenge([18, 20, 0, 2])
= laenge([18, 20, 0 + 18 · 0.13, 2 + 20 · 0.15]) / (20 − 18 + 0 + 2)
= laenge([18, 20, 2.34, 5]) / 4
= 9.34 / 4
= 2.34
P1 P3 ε3

∆3(P3, T ↑
3 (P3)) = laenge(T ↑
3
([5, 6, 0, 0])) / laenge([5, 6, 0, 0])
= laenge([5, 6, 0, 0] ⊗ [1, 1, ε3, ε3/(1 − ε3)) / laenge([5, 6, 0, 0])
= laenge([5, 6, 5 · ε3, 6 · ε3/(1 − ε3)]) / 1
= 1 + 5 · ε3 + 6 · ε3/(1 − ε3)
=⇒ ε3 ≈ 0.11
T ↑
1 (P1) T ↑
1 (P1) = [18, 20, 2.34, 5]
P3 ε3 = 0.11 T ↑
3 (P3) =
[5, 6, 0.55, 0.74]
Q1 = T ↑
1 (P1) ∧ P3 Q2 = P1 ∧ T ↑
3 (P3)
Q 1
↑(1)
T (P ) 1 1
Q’
P Ʌ P
1 3
{0}
Q 2
Ʌ P {0} ↑(1)
P 3 1 Ʌ T (P ) {0}
3 3
P1 ∧ P3
Q ′
Cl[M 2 1 ] Cl[M 2 3 ] ε1
ε3 M 2 1
M 2 1 = [1, 1, 2 · ε1 − ε 2 1, (2 · ε1 − ε 2 1)/(1 − ε1) 2 ] = [1, 1, 0.26 − 0.13 2 , (0.26 − 0.13 2 )/0.87 2 ]
≈ [1, 1, 0.24, 0.32]

M 2 3
M 2 3 = [1, 1, 2 · ε3 − ε 2 3, (2 · ε3 − ε 2 3)/(1 − ε3) 2 ] = [1, 1, 0.22 − 0.11 2 , (0.22 − 0.11 2 )/0.89 2 ]
≈ [1, 1, 0.21, 0.26].
ε1
M i 1 , i > 0 V M 2 1
M 2 3 supp(M 2 3 ) = [0.79, 1.26] ⊂ V =
[( √ 5 − 1)/2, ( √ 5 + 1)/2] ≈ [0.61, 1.61] M 2 3
P1 P3
P1
P3
T ↑(2)
1 (P1) = P1 ⊗ M 2 1 = [18, 20, 0, 2] ⊗ [1, 1, 0.24, 0.32]
= [18, 20, 0 + 18 · 0.24, 2 + 20 · 0.32 = [18, 20, 4.32, 8.4]
T ↑(2)
3 (P3) = P3 ⊗ M 2 3 = [5, 6, 0, 0] ⊗ [1, 1, 0.21, 0.26]
= [5, 6, 0 + 5 · 0.21, 0 + 6 · 0.26 = [5, 6, 1.05, 1.56].
Q ′
Q3 = T ↑(2)
1 (P1) ∧ P3 = [18, 20, 4.32, 8.4] ∧ [5, 6, 0, 0]
Q4 = T ↑
1 (P1) ∧ T ↑
3 (P3) = [18, 20, 2.34, 5] ∧ [5, 6, 0.55, 0.74]
Q5 = P1 ∧ T ↑(2)
3 (P3) = [18, 20, 0, 2] ∧ [5, 6, 1.05, 1.56]
Q4 Q5
Q3
Q ′
Q
Q ∗ = T ↑(2) (Q) = T ↑(2)
1 (P1) ∧ P2 ∧ P3
= ( [18, 20, 4.32, 8.4]) ∧ (∅ [0.7, 1.7, 0, 0.6]) ∧ ( [5, 6, 0, 0]).

Q 3
↑(2)
T (P ) 1 1
Ʌ P 3
↑(1)
T (P ) 1 1
{1}
Q 1
P Ʌ P
1 3
Ʌ P {0} ↑(1)
P 3 1 Ʌ T (P ) {0}
3 3
↑(1)
T (P ) 1 1
Q’
Q 4
{0}
Q 2
Q 5
↑(1)
↑(2)
Ʌ T (P ) {0} P1 Ʌ T (P ) {0}
3 3
3 3
P1 ∧ P3
P1
Q ∗ P1 P3
µQ ∗ µP1 µP2 µP3 µ T ↑
1 (P1)
µ T ↑
3 (P3)
µ ↑(2)
T1 (P1)
µ T ↑(2)
3 (P3)
∅
Q
Q ∗
Q ∗ = ∅ Q∗

Q ∗

α
α
α

P
P
T ↓
T ↑
T ↓ ∀x ∈
dom(A), µ T ↓ (P )(x) ≤ µP (x) A P
dom(A) A
T ↓ P core(T ↓ (P )) ⊂ core(P )
T ↓ P T ↓ (P ) = [A ′ , B ′ , a, b]
P = [A, B, a, b] A ′ − A < a B − B ′ < b
a, b > 0
E[Z]
Z P
Z Z = [−δ, δ, 0, 0]
P ′
1
0
µ
P ′ = T ↓ (P ) = P ΘZ
δ 0 δ
Z
R

P Θ
P =
[A, B, a, b]
P ′ = T ↓ (P ) = [A, B, a, b]Θ[−δ, δ, 0, 0] = [A + δ, B − δ, a, b] = [A ′ , B ′ , a, b].
δ Z 0 < δ ≤ (B − A)/2
P ′
A ′ ≤ B ′ Θ
E[Z] Θ
P δ
P
δ
1
0
µ
A-a
P’
A’-a A A’ B’ B B’+b
δ δ
P P ′
Q ∗ Q ∗ Q
Q Q ∗
Q ⊆ Q
Q = Q0 = P P
n ∗ Q
i 0 < i ≤ n Qi
P
B+b
Qi = T ↓(i) (P ) = P Θi · Z = [A + i · δ, B − i · δ, a, b].
U

A + i · δ ≥ B − i · δ
δ A+i·δ ≥ B −i·δ
n
⌊x⌋ = max
y∈Z,y≤x (y)
n = ⌊(B − A)/(2 · δ)⌋,
(B−A)/2 P = [A, B, a, b]
Z
1
µ
0 A-a A B B+b
1
0
µ
A B
1
0
µ
A B B+b
P = [A, B, a, b]
δ < min(a, b)
P ′
Qi ∗ Qi−1
a = 0 b = 0
∗
Qi
Z = [−δneu, δneu, 0, 0]
δneu < δalt
P
P = [A, B, 0, b] P = [A, B, a, 0]

δ ∈ R
max ∈ N
Z = [−δ, δ, 0, 0]
Q0
∗ berechne∗ (Qi)
| ∗ | i ≤ (B − A)/(2 · δ))
Qi P Θi · Z
berechne ∗ (Qi)
Qi
1
0
µ
Q P
A B’ δ B
B’+b
P’
P
B+b
U
1
0
µ
P
P’
175 185 195
205
250
a, b = 0 Z Z = [−δ, δ, 0, 0]
Z
Z = [0, δ, 0, 0] Z = [−δ, 0, 0, 0] δ < b δ < a
P = [A, B, 0, b] P ′ = [A, B − δ, 0, b]
P
P = [195, 250, 20, 0] Z = [10, 0, 0, 0]
P
U

P = [A, B, 0, 0]
Z = [−δ, δ, 0, 0]
P P ′ = [A + δ, B − δ, 0, 0]
Z = [−δ, δ, γ, γ]
γ = δ+ε ε γ
A B
P ′
P ′ = [A + δ, B − δ, a + δ, b + δ]
Q = P
Q = P1 ∧ · · · ∧ Pn
P1 Pn
Q = Q0 = P1 ∧ · · · ∧ Pn T ↓ Q1
Qi = T ↓ (Qi−1) = T ↓(i)
1 (P1) ∧ · · · ∧ T ↓(i)
n (Pn).
T ↓
i

Pi
Qi Q0 Qj
i < j
∆(Pi, T ↓
i (Pi)) = laenge(core(T ↓
i (Pi))) / laenge(core(Pi))
laenge([x, y]) = y − x x, y ∈ R Pi
T ↓
i (Pi)
laenge(core(Pi)) = 0
∆(Pi, T ↓
i (Pi)) i = 1, . . . , n ∆(P1, T ↓
1 (P1)) =
· · · = ∆(Pn, T ↓ n(Pn))
≺
Qi ≺ Qj ⇐⇒
n
k=1
anzahl(T ↓
k in Qi) <
n
k=1
anzahl(T ↓
k
in Qj).
Qi Qj Q
Q = P1 ∧ · · · ∧ Pn Pi
1 ≤ i ≤ n Zi
Zj j = i
Q = P1 ∧· · ·∧Pn
Q
Pi 0 ≤ i ≤ n

P1 ∧ · · · ∧ Pn n
Pfix ∈ {P1, . . . , Pn} −
δfix ∈ R Pfix
Zfix = [−δfix, δfix, 0, 0]
max ∈ N
≤
=
Zi −
Q0
∗ berechne∗ (Qi)
| ∗ |
≤ (Bj − Aj) / 2 · δj
T ↓(i)
(Pj) = PjΘi · Zj
Qi = T ↓(i)
1 (P1) ∧ · · · ∧ T ↓(i)
n (Pn)
∗
berechne∗ (Qi)
j
− Zfix
Qi
P1 Pn P∗
P∗
P∗
P∗
mi ∗
Q(Pi) ∗
Q(Pi)
Pi 0 ≤ i ≤ n ∗ Q

mi
mi = mittelwert(Pi) :=
[A,B,a,b]∈ ∗
Q(P i )
(A + B/2) /
∗
Q(Pi)
Pi =
[Ai, Bi, ai, bi] mi
core(Pi)
di = min(mi − Ai, Bi − mi) / laenge(core(Pi)).
P∗ di
P∗ = Pi i = min (j)
∀k dj≤dk
i P∗
Zi
Qi = T ↓ (Qi−1) = T ↓(k1)
1 (P1) ∧ · · · ∧ T ↓(kn)
n (Pn), k1, . . . , kn ≥ 0
k1
kn i P∗
k∗ ∗
Qi
.

P1 ∧ · · · ∧ Pn n
δi ∈ R Pi
Zi = [−δi, δi, 0, 0]
max ∈ N
k1 = · · · = kn = 0
T ↓(0)
j (Pj) = Pj
Q0
∗ berechne∗ (Qi)
|
∗ |
≤
mj = mittelwert(Pj)
dj = min(mj − Aj, Bj − mj) / laenge(core(Pj))
Pj = [Aj, Bj, aj, bj]
≤
dj < dmin
P∗
kj kj
T ↓(k
min)
min
(Pmin) = PminΘkmin · Zmin
Qi = T ↓(k1)
(P1) ∧ · · · ∧ T ↓(kn)
(Pn)
1
∗
berechne∗ (Qi)
n
Qi
Q Q = ¬P
P ¬ P
M = ¬M M µ M (x) = 1 − µM(x)
x ∈ M ¬P
¬P
P T ↓
¬P T ↑ P
T ↓ (¬P ) = T ↑ (P )
Z

Q = ¬P
¬P
P
T ↑ (¬P ) = T ↓ (P ).
∗ | ∗| > 1

Q
Q = ( [20, 25, 5, 2] ) ∧ ( [30, 40, 6, 3] ).
P1
1
0
µ
P
1
15 20 25
27
Alter
1
0
µ
P
2
P2
24 30
40
P1 P2
Q
Q
43
Gehalt
∗
Q
∗
Q
= 3 > 1

Z1 Z1 = [−1, 1, 0, 0]
∆1(P1, T ↓
1 (P1)) = ∆2(P2, T ↓
2 (P2))
∆i(Pi, T ↓
i (Pi)) = laenge(core(T ↓
i (Pi))) / laenge(core(Pi)).
∆1
∆1(P1, T ↓
1 (P1)) = laenge(core(T ↓
1
([20, 25, 5, 2]))) / laenge(core([20, 25, 5, 2]))
= laenge(core([20, 25, 5, 2]Θ[−1, 1, 0, 0]))/laenge([20, 25])
= laenge(core([21, 24, 5, 2]))/laenge([20, 25])
= 3/5.
Z2 = [−δ2, δ2, 0, 0] P2
∆2(P2, T ↓
2 (P2)) = laenge(core(T ↓
2
([30, 40, 6, 3]))) / laenge(core([30, 40, 6, 3]))
= laenge(core([30 + δ2, 40 − δ2, 6, 3])) / laenge([30, 40]))
= (40 − δ2 − 30 − δ2)/40 − 30
= (10 − 2 · δ2)/10 = (5 − δ2)/5
=⇒ δ2 = 2
Z2 = [−2, 2, 0, 0]
Q1 = T ↓ (Q) = T ↓
1 (P1) ∧ T ↓
2 (P2)
= ( [21, 24, 5, 2]) ∧ ( [32, 38, 6, 3]).
Q1
∗
Q1
∗ Q1
= 2 > 1

Q1
Z1 Z2
Q2
Q2 = T ↓(2) (Q) = T ↓(2)
1
(P1) ∧ T ↓(2)
2 (P2)
= ( [22, 23, 5, 2]) ∧ ( [34, 36, 6, 3]).
Q2
Q

1
0
µ
a b c a’ d
P 1
1
a b c d d’
1 1 1 2 1 2
2 2
2 2
P
2
P’
2

P1 = [a1, b1, c1, d1] P2 = [a2, b2, c2, d2] a1, b1, c1, d1 < a2
P1 P2
P1 P2
α = [aα, bα, cα, dα] β = [aβ, bβ, cβ, dβ]
⎧
⎪⎨
1, cα ≥ bβ
dα−aβ
ABSP OSGR(α, β) =
(dα−aβ)+(bβ−cα)
⎪⎩
, cα < bβ dα > aβ
0,
β
aβ ≤ dα aβ aβ ≤ dα
P1 P2
P2
⎧
⎪⎨
1, cα ≥ dβ
dα−cβ
RELP OSGR(α, β) =
(dα−cβ)+(dβ−cα)
⎪⎩
, cα < dβ dα > cβ
0,
aβ dβ
cβ dα
⎧
⎪⎨
1, aα ≥ dβ
bα−cβ
ABSNECGR(α, β) =
(bα−cβ)+(dβ−aα)
⎪⎩
, aα < dβ bα > cβ
0,
cβ

P1 = [a1, b1, c1, d1]
P2 = [a2, b2, c2, d2] P1 P2
1
0
µ
P 1
a a b a’ b c d c’ c d’ d
1 2 b’ 1 2 2 1 2
1 2
2 2 2
c1 ≥ b2
P2 c1 < b2
P1 P2
b1 ≥ c2
P2
1
0
µ
P’
2
P
2
a 2 a’ b c’ 2 2 2 2 c2 a d’ d b c d
b’ 1 2 2 1
1
1
P 1
P’
2
P
2

P2
α β S1(α, β) 1
α β P aT N3sup(α, β)
α β P aT N2rel(α, β)
α β InSN2rel(α, β)
α β P aSN3inf (α, β)
α β NK3(α, β)
α β F GEQ(α, β)
α β F GT (α, β)
α β NF GEQ(α, β)
α β NF GT (α, β)
α β NK3(β, α)
α β F GEQ(β, α)
α β NF GEQ(β, α)
α β F GT (β, α)
α β NF GT (β, α)
P2
P1 P2 = 1 ⇐⇒ P1 = [a, b, c, d] = P2 P2
P2

ν θ
δ
δ
FP

ν θ
δ
δ
FP

δ

δ

δ

δ

δ

δ

δ

+

+

+

+

+

+

+

+

−

−
−

−
−