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5 Operations andAggregations ofFuzzy Sets
Fuzzy Systems EngineeringToward Human-Centric Computing
5.1 Standard operations on sets and fuzzy sets
5.2 Generic requirements for operations on fuzzy se ts
5.3 Triangular norms
5.4 Triangular conorms
5.5 Triangular norms as a general category of logic al operations
5.6 Aggregation operations
5.7 Fuzzy measure and integral
5.8 Negations
Contents
Pedrycz and Gomide, FSE 2007
5.1 Standard operations on sets and fuzzy sets
Pedrycz and Gomide, FSE 2007
Intersection of sets
Pedrycz and Gomide, FSE 2007
A = x∈R| 1 ≤ x ≤ 3
B = x∈R| 2 ≤ x ≤ 4 (A∩B)(x) = min [A(x), B(x)] ∀x∈X
A∩B: x∈R| 2 ≤ x ≤ 3
1.0
x
A(x)
A B1.0
x
A(x)
BA
A∩B
1 2 3 4 1 2 3 4
Pedrycz and Gomide, FSE 2007
Union of sets
A = x∈R| 1 ≤ x ≤ 3
B = x∈R| 2 ≤ x ≤ 4 (A∪B)(x) = max [A(x), B(x)] ∀x∈X
A∪B: x∈R| 1≤ x ≤ 4
1.0
x
A(x)
A B1.0
x
A(x)
BA
A∪B
1 2 3 4 1 2 3 4
Complement of sets
A = x∈R| 1 ≤ x ≤ 3
1.0
x
A(x)
A1.0
x1 2 3 4 1 2 3 4
A(x)
A(x) = x∈R| x < 1 , x > 3
A(x) = 1 –A(x) ∀x∈X
A
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
Intersection of fuzzy sets
(A∩B)(x) = min [A(x), B(x)] ∀x∈X Standard intersection
1.0
x
A(x)A B
1 2 3 4
1.0
x
A(x)A B
1 2 3 4
A∩B
Pedrycz and Gomide, FSE 2007
Union of fuzzy sets
(A∪B)(x) = max [A(x), B(x)] ∀x∈X Standard union
1.0
x
A(x)A B
1 2 3 4
1.0
x
A(x)A B
1 2 3 4
A∪B
Pedrycz and Gomide, FSE 2007
Complement of fuzzy sets
Standard complement
1.0
x
A(x)A
1 2 3 4
1.0
x
A
1 2 3 4
AA(x)
A(x) = 1 –A(x) ∀x∈X
1 CommutativityA ∪ B = B ∪ A
A ∩ B = B ∩ A
2 AssociativityA ∪ (B ∪ C) = (A ∪ B) ∪ C
A ∩ (B ∩ C) = (A ∩ B) ∩ C
3 DistributivityA ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
4 IdempotencyA ∪ A = A
A ∩ A = A
5 Boundary ConditionsA ∪ φ = A and A ∪ X = X
A ∩ φ = φ and A ∩ X = A
6 Involution
7 Transitivity if A ⊂ B and B ⊂ C then A ⊂ C
A = A
Basic properties of sets and fuzzy sets
Pedrycz and Gomide, FSE 2007
Sets Fuzzy sets
8-Noncontradiction
9-Excluded middle A∪A = X A∪A ≠ X
A∩A = φ A∩A ≠ φ
Noncontradiction and excluded middle for standard operations
Pedrycz and Gomide, FSE 2007
Geometric view of standard operations
1.0
1.0
x2
x1
A
F(X)
∅
x1, x2
A∪A
AA∩A
Pedrycz and Gomide, FSE 2007
1.0
1.0
x2
x1
F(X)
∅
x1, x2
A=A
1.0
x2x1
0.5
X
A(x)
Example
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
5.2 Generic requirements foroperations on fuzzy sets
Set operation is a binary operator
Pedrycz and Gomide, FSE 2007
[0,1] ×[0,1] → [0,1]
Requirements:
– commutativity– associativity– identity
Identity:
– its form depends on the operation
Pedrycz and Gomide, FSE 2007
5.3 Triangular norms
Definition
t : [0,1] ×[0,1] → [0,1]
• Commutativity: a t b = b t a
• Associativity: a t (b t c) = (a t b) t c
• Monotonicity: if b ≤ c then a t b ≤ a t c
• Boundary conditions: a t 1 = aa t 0 = 0
Pedrycz and Gomide, FSE 2007
Examples
(a) (b)
a tm b = min(a,b) a tp b = ab
minimum product
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
a tl b = max(a+b – 1,0)
(c) (d)
Lukasiewicz drasticproduct
==
=otherwise0
1if
1if
d ab
ba
bat
),min( baatbbatd ≤≤
aata ≤
lower and upper bounds
Archimedean (if t is continuous)
0=na nilpotent (e.g. Lukasiewicz)
a t a = a2, … , an-1 t a = an
Pedrycz and Gomide, FSE 2007
Constructors of t-norms
Monotonic function transformation
Additive and multiplicative generators
Ordinal sums
Pedrycz and Gomide, FSE 2007
Monotonic function transformation
If h: [0,1] → [0,1] is a strictly increasing bijection
then th(a,b) = h-1(t(h(a),h(b))) is a t-norm
h is a scaling transformation
Obs: h bijective means both, h injective (one-to-one)and h surjective (onto)
Pedrycz and Gomide, FSE 2007
(a)
(b)
Examples
h = x2, x∈[0,1]
minimum
product
t th
Pedrycz and Gomide, FSE 2007
(c)
(d)
Lukasiewicz
drastic product
h = x2, x∈[0,1]
t th
Pedrycz and Gomide, FSE 2007
Additive generators
f: [0,1] → [0, ∞), f(1) = 0
– continuous – strictly decreasing
a tf b = f –1(f(a) + f(b)) ⇔ is a Archimedean t-norm
Pedrycz and Gomide, FSE 2007
1.0 x
R+
f(x)
f(a)
f(b)
f(a) + f(b)
a ba tf b
Additive generators of t-norms
R+ ≡ [0, ∞),
Pedrycz and Gomide, FSE 2007
Example
f = – log(x)
f(a) + f(b) = – log(a) – log(b)
– (log(a) + log(b))
– log(ab)
f- –1(f(a) + f(b)) = elog(ab) = ab
a tf b = ab (Archimedean t-norm)
Pedrycz and Gomide, FSE 2007
Multiplicative generators
g: [0,1] → [0, 1], g(1) = 1
– continuous – strictly increasing
a tg b = g –1(g(a)g(b)) ⇔ is a Archimedean t-norm
Pedrycz and Gomide, FSE 2007
Multiplicative generators of t-norms
1.0 x
g(x)
g(a)
g(b)
g(a)g(b)
a ba tg b
1.0
Pedrycz and Gomide, FSE 2007
Example
g = x2
a tf b = ab (Archimedean t-norm)
g = e–f(x)
multiplicative and additive generators → same t-norm
Pedrycz and Gomide, FSE 2007
Ordinal sums
to: [0,1] → [0, 1] denoted to = (<αk,βk, tk>, k∈K)
βα∈
α−βα−
α−βα−α−β+α=τ
otherwise)min(
][if)()(o
b,a
,b,ab
,a
t,I,b,at kk
kk
k
kk
kkkkk
I = [αk,βk], k∈K
nonempty, countable family pairwise disjoint subintervals of [0,1]
τ = tk, k∈K
family of t-normsPedrycz and Gomide, FSE 2007
Example
∈−++∈−−+
=τotherwise)min(
]7050[if)021max(50
]4020[if)20)(20(520
)(o
b,a
.,.b,a,.ba.
.,.b,a.b.a.
,I,b,at
I = [0.2, 0.4], [0.5, 0.7] K =1,2
τ = tp, tl, t1 = tp, t2 = tl
Pedrycz and Gomide, FSE 2007
∈−++∈−−+
=τotherwise)min(
]7050[if)021max(50
]4020[if)20)(20(520
)(o
b,a
.,.b,a,.ba.
.,.b,a.b.a.
,I,b,at
Pedrycz and Gomide, FSE 2007
∈−++∈−−+
=τotherwise)min(
]7050[if)021max(50
]4020[if)20)(20(520
)(o
b,a
.,.b,a,.ba.
.,.b,a.b.a.
,I,b,at
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
5.4 Triangular conorms
Pedrycz and Gomide, FSE 2007
Definition
s : [0,1] ×[0,1] → [0,1]
• Commutativity: a s b = b s a
• Associativity: a s (b s c) = (a s b) s c
• Monotonicity: if b ≤ c then a s b ≤ a s c
• Boundary conditions: a s 1 = 1a s 0 = a
Pedrycz and Gomide, FSE 2007
Examples
a sm b = max(a,b) a sp b = a + b – ab
maximum probabilisticsum
(b) (a)
Pedrycz and Gomide, FSE 2007
a sl b = max(a+b, 1)
Lukasiewicz drasticsum
==
=otherwise1
0if
0if
d ab
ba
bas
(c) (d)
Pedrycz and Gomide, FSE 2007
basasbb,a d)max( ≤≤
aasa ≥
lower and upper bounds
Archimedean (if t is continuous)
1=na nilpotent (e.g. Lukasiewicz)
a s a = a2, … , an-1 s a = an
Pedrycz and Gomide, FSE 2007
Dual norms and De Morgan laws
a s b = 1 – (1 –a) t (1 –b)
a t b = 1 – (1 –a) s (1 –b)
(1 –a) t (1 –b) = 1 –a s b
(1 –a) s (1 –b) = 1 –a t b
BABA
BABA
∩=∪∪=∩
Dual triangular norms
De Morgan
Pedrycz and Gomide, FSE 2007
Constructors of s-norms
Monotonic function transformation
Additive and multiplicative generators
Ordinal sums
Pedrycz and Gomide, FSE 2007
Monotonic function transformation
If h: [0,1] → [0,1] is a strictly increasing bijection
then sh(a,b) = h-1(s(h(a),s(b))) is a t-norm
h is a scaling transformation
Pedrycz and Gomide, FSE 2007
Examples
h = x2, x∈[0,1]
maximum
probabilisticsum
s sh
(a)
(b)
Pedrycz and Gomide, FSE 2007
Lukasiewicz
drastic sum
h = x2, x∈[0,1]
s sh
(c)
(d)
Pedrycz and Gomide, FSE 2007
Additive generators
f: [0,1] → [0, ∞), f(0) = 0
– continuous – strictly increasing
a sf b = f –1(f(a) + f(b)) ⇔ is a Archimedean t-conorm
Additive generators of t-conorms
1.0 x
R+
f(x)
f(a)f(b)
f(a) + f(b)
a b a sf b
Pedrycz and Gomide, FSE 2007
Example
f = – log(1 –x)
f(a) + f(b) = – log(1 –a) – log(1 –b)
– (log(1 –a) + log(1 –b))
– log(1 –a)(1 –b)
f- –1(f(a) + f(b)) = 1 – elog(1 –a)(1 –b) = a + b – ab
a sf b = a + b – ab (Archimedean t-conorm)
Pedrycz and Gomide, FSE 2007
Multiplicative generators
g: [0,1] → [0, 1], g(0) = 1
– continuous – strictly decreasing
a sg b = g –1(g(a)g(b)) ⇔ is a Archimedean t-norm
Pedrycz and Gomide, FSE 2007
1.0 x
R+
g(x)g(a)
g(b)
g(a)g(b)
a b a sg b
Multiplicative generators of t-conorms
R+ ≡ [0, ∞),Pedrycz and Gomide, FSE 2007
Example
g = 1 –x
a sg b = a + b – ab (Archimedean t-norm)
g = e–f(x)
multiplicative and additive generators → same t-conorm
Pedrycz and Gomide, FSE 2007
Ordinal sums
so: [0,1] → [0, 1] denoted so = (<αk,βk, sk>, k∈K)
βα∈
α−βα−
α−βα−α−β+α=σ
otherwise)max(
][if)()(o
b,a
,b,ab
,a
s,I,b,as kk
kk
k
kk
kkkkk
I = [αk,βk], k∈K
nonempty, countable family pairwise disjoint subintervals of [0,1]
σ = sk, k∈K
family of t-conormsPedrycz and Gomide, FSE 2007
Example
∈−+−+∈−+−+
=σotherwise)max(
]7050[if)1)020(5)20min(5(2050
]4020[if)20)(20(5)20()20(20
)(o
b,a
.,.b,a,,.b.a..
.,.b,a.b-.a--.b.a.
,I,b,as
I = [0.2, 0.4], [0.5, 0.7] K =1,2
σ = sp, sl, s1 = sp, s2 = sl
Pedrycz and Gomide, FSE 2007
∈−+−+∈−+−+
=σotherwise)max(
]7050[if)1)020(5)20min(5(2050
]4020[if)20)(20(5)20()20(20
)(o
b,a
.,.b,a,,.b.a..
.,.b,a.b-.a--.b.a.
,I,b,as
Pedrycz and Gomide, FSE 2007
∈−+−+∈−+−+
=σotherwise)max(
]7050[if)1)020(5)20min(5(2050
]4020[if)20)(20(5)20()20(20
)(o
b,a
.,.b,a,,.b.a..
.,.b,a.b-.a--.b.a.
,I,b,as
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
5.5 Triangular norms as generalcategory of logical operators
Pedrycz and Gomide, FSE 2007
Motivation
Fuzzy propositions involves linguistic statements:
– temperature is low and humidity is high– velocity is high or noise level is low
Logical operations:
– and (∧)– or (∨)
Pedrycz and Gomide, FSE 2007
L = P, Q, .... P, Q, ... atomic statements
truth: L → [0, 1] p, q,... ∈ [0, 1]
truth (P and Q) = truth (P ∧ Q) → p ∧ q = p t q
truth (P or Q) = truth (P ∨ Q) → p ∨ q = p s q
Truth value assignment
Pedrycz and Gomide, FSE 2007
p q min(p, q) max(p, q) pq p + q – pq
1 1 1 1 1 1
1 0 0 1 0 1
0 1 0 1 0 1
0 0 0 0 0 0
0.2 0.5 0.2 0.5 0.1 0.6
0.5 0.8 0.5 0.8 0.4 0.9
0.8 0.7 0.7 0.8 0.56 0.94
Examples
a ϕ b ≡ a ⇒ b
a ϕ b = sup c ∈ [0, 1] | a t c ≤ b ∀a,b ∈ [0,1]
a b a ⇒ b a ϕ b
0 0 1 1
0 1 1 1
1 0 0 0
1 1 1 1
ϕ operator or Boolean values of its arguments
Implication induced by a t-norm
residuation
Pedrycz and Gomide, FSE 2007
5.6 Aggregation operations
Pedrycz and Gomide, FSE 2007
Pedrycz and Gomide, FSE 2007
Definition
g : [0,1]n → [0,1]
1. Monotonicity:
g (x1, x2,..., xn) ≥ g (y1, y2,.., yn) if xi ≥ yi, i = 1,.., n
2. Boundary conditions:
g (0, 0,..., 0) = 0g (1, 1,..., 1) = 1
Pedrycz and Gomide, FSE 2007
1. Neutral element (e):
g (x1, x2,..., xi-1, e, xi+1,..., xn) = g (x1, x2,..., xi-1, e, xi+1,..., xn) n ≥ 2
2. Annihilator (l ):
g (x1, x2,..., xi-1, l, xi+1,..., xn) = l
Observation: Annihilator ≡ absorbing element
Averaging operations
Pedrycz and Gomide, FSE 2007
0)(1
)(1
21 ≠∈∑==
p,p,xn
x,,x,xg pn
i
pin RK
∑
=−=
∏=→
∑==
=
=
=
n
ii
n
nn
iin
n
iin
x/
nx,,x,xgp
xx,,x,xgp
xn
x,,x,xgp
1
21
121
121
1)(1
)(0
1)(1
K
K
K arithmetic mean
geometric mean
harmonic mean
Pedrycz and Gomide, FSE 2007
p → – ∝ g (x1, x2,..., xn) = min (x1, x2,..., xn)
p → ∝ g (x1, x2,..., xn) = max (x1, x2,..., xn)
Bounds
min (x1, x2,..., xn) ≤ g (x1, x2,..., xn) ≤ max (x1, x2,..., xn)
Pedrycz and Gomide, FSE 2007
Examples
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
(b) Arithmetic mean of A and B
A B
Arithmetic mean
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
(d) Geometric mean of A and B
A B
Geometric mean
Pedrycz and Gomide, FSE 2007
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
(f) Harmonic mean of A and B
A B
Harmonic mean
Pedrycz and Gomide, FSE 2007
Ordered Weighted Averaging (OWA)
∑==
n
iii xAwA
1)(),OWA( w
Σwi = 1, wi ∈ [0, 1]
A (x1) ≤ A(x2) ≤... ≤ A( xn)
Pedrycz and Gomide, FSE 2007
1. w = [1, 0,...,0] OWA(A,w) = min (A(x1), A(x2),..., A(xn))
2. w = [0, 0,...,1] OWA(A,w) = max (A(x1), A(x2),..., A(xn))
3. w = [1/n, 1/n,...,1/n] OWA(A,w) = arithmetic mean
min (A(x1), A(x2),..., A(xn)) ≤ OWA(A,w) ≤ max (A(x1), A(x2),..., A(xn))
Examples
Pedrycz and Gomide, FSE 2007
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
(h) Owa of A and B, w1 = 0.8, w2 = 0.2
A B
w = [0.8, 0.2]
Examples
Pedrycz and Gomide, FSE 2007
0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
(f) Owa of A and B, w1 = 0.2, w2 = 0.8
A B
w = [0.2, 0.8]
Pedrycz and Gomide, FSE 2007
Uninorms
u : [0,1] ×[0,1] → [0,1]
Commutativity: a u b = b u a
Associativity: a u (b u c) = (a u b) u c
Monotonicity: if b ≤ c then a u b ≤ a u c
Identity: a u e = a ∀a ∈ [0, 1]
e ∈ [0, 1], e = 1 u is a t-norm, e = 0 u is a t-conorm
Pedrycz and Gomide, FSE 2007
conormtandnormtareand1
))1(())1((
)()(
−−−
−−+−+=
=
uu
u
u
ste
ebeeuaeebas
e
ebueabat
Results on uninorms
1. tu, su: [0, 1] × [0, 1] → [0, 1] such that ∀e ∈ [0, 1]
Pedrycz and Gomide, FSE 2007
2. If a ≤ e ≤ b or a ≥ e ≥ b then
),max(),min(
thenor If
baaubba
beabea
≤≤≥≥≤≤
Pedrycz and Gomide, FSE 2007
≤≤≤≤
=
≤≤≤≤
=
≤≤
otherwise)max(
1if1
0if)min(
otherwise)min(
1if)max(
0if0
b,a
b,ae
eb,ab,a
bau
b,a
b,aeb,a
eb,a
bau
bauaubbau
s
w
sw
3. For any u with e ∈ [0, 1]
Pedrycz and Gomide, FSE 2007
≤≤−−
−−−+
≤≤
=
=
≤≤−−
−−−+
≤≤
=
=
otherwise)max(
1if1
(1
)(1(
0if)()(
then1)10(if
otherwise)min(
1if1
(1
)(1(
0if)()(
then0)10(if
b,a
b,ae)e
ebs)
e
eaee
eb,ae
bt
e
ae
bau
u
b,a
b,ae)e
ebs)
e
eaee
eb,ae
bt
e
ae
bau
u
d
c
4. Conjunctive and disjunctive uninorm
Pedrycz and Gomide, FSE 2007
Conjunctive uninorm
e = 0.5
Pedrycz and Gomide, FSE 2007
e = 0.5
Disjunctive uninorm
Pedrycz and Gomide, FSE 2007
5. Almost continuous Archimedian uninorms
a u a < a for 0 < a < e
a u a > a for e < a < 1
6. Additive and multiplicative generators of almost continuous uninorms
a uf b = f –1(f(a) + f(b))
a ug b = g –1(g(a)g(b))
g(x) = e–f(x)
f strictly increasing
g strictly decreasing
Pedrycz and Gomide, FSE 2007
7. Ordinal sum
≥∉
ι∈
α−βα−
α−βα−α−β+α
ι∈
α−βα−
α−βα−α−β+α
=στ
otherwise)min(and
][if)max(
][if)(
][if)(
)( 2
1
co
b,aeb,a
,βαa,bb,a
b,ab
,a
s
b,ab
,a
t
,,ι,b,au
kk
kk
k
kk
kkkkk
kk
k
kk
kkkkk
l = [ αk, βk], k∈Kl1 = [ αk, βk] ∈ l | βk ≤ el2 = [ αk, βk] ∈ l | αk ≥ eτ = tk, k∈K, σ = sk, k∈K
Pedrycz and Gomide, FSE 2007
≤∉
ι∈
α−βα−
α−βα−α−β+α
ι∈
α−βα−
α−βα−α−β+α
=στ
otherwise)max(and
][if)min(
][if)(
][if)(
)( 2
1
co
b,aeb,a
,βαa,bb,a
b,ab
,a
s
b,ab
,a
t
,,ι,b,au
kk
kk
k
kk
kkkkk
kk
k
kk
kkkkk
Pedrycz and Gomide, FSE 2007
Nullnorms
v : [0,1] ×[0,1] → [0,1]
Commutativity: a v b = b v a
Associativity: a v (b v c) = (a v b) v c
Monotonicity: if b ≤ c then a v b ≤ a v c
Absorbing element: a v e = e ∀a ∈ [0, 1]
Boundary conditions: a v 0 = a ∀a ∈ [0, e]a v 1 = a ∀a ∈ [e, 1]
Pedrycz and Gomide, FSE 2007
e ∈ [0, 1]
v behaves as a t-norm in [0, e]×[0, e]
v behaves as a t-conorm in [e, 1]×[e, 1]
v = e in the rest of the unit square
e
ebeevaeebat
e
ebveabas
v
v
−−−+−+=
=
1))1(())1((
)()(
Pedrycz and Gomide, FSE 2007
e = 0.5, t-norm = min, t-conorm = max
Example
Pedrycz and Gomide, FSE 2007
Symmetric sums
σs(a1, a2,...., an) = 1 –σs(1 –a1, 1 –a2,...., 1 –an)
1
21
2121 )(
)11(1)(
−
−−+=σn
nns a,,a,af
a,,a,afa,,a,a
K
KK
f increasing, continuous
f (0,0,...,0) = 0
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0 1 2 3 4 5 6 7 8 0
0.2
0.4
0.6
0.8
1
1.2
x
Symmetric sum of A and B
A B
Example
f (a,b) = a2 + b2
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Compensatory operations
a b = (a t b)1 –γ (a s b)γ
a b = (1 –γ)(a t b) + γ(a t b)
compensatory product
compensatory sum
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(a) (b)
Example
γ = 0.5, t-norm = min, t-conorm = max
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5.7 Fuzzy measure andintegral
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Fuzzy measure
g : Ω → [0,1]
• Boundary conditions: g(∅) = 0g(X) = 1
• Monotonicity: if A ⊂ B then g(A) ≤ g(B)
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λλλλ–fuzzy measure
g(A∪B) = g(A) + g(B) + λg(A) g(B), λ > − 1
• λ = 0 g(A∪B) = g(A) + g(B) additive
• λ > 0 g(A∪B) ≥ g(A) + g(B) super-additive
• λ < 0 g(A∪B) ≤ g(A) + g(B) sub-additive
Fuzzy integral
h : X → [0,1] Ω measurable
fuzzy integral of h with respect to g over A
∫ ∩α= α∈α
A,
HAg,gxh )](min[sup)()(]10[
o
Hα = x | h(x) ≥ α
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X = x1, x2,....,xn
h(x1) ≥ h(x2) ≥ ..... ≥ h(xn)
A1 = x1), A2 = x1, x2), ..., An = x1, x2,..., xn = X
∫ ==A ii
,n,iAg,xhgxh )]()(min[max)()(
1Ko
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1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
xi
Fuzzy integra l
h(xi)
g(Ai)
Example
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Choquet integral
)()]()([)( 11
iin
ii AgxhxhgfCh +
=−∫ ∑=o
h(xn+1) = 0
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5.8 Negations
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Definition
N : [0,1] → [0,1]
1. Monotonicity:
N is nonincreasing
2. Boundary conditions:
N (0) = 1N (1) = 0
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3. Continuity:
N is a continuous function
4. Involution:
N (N(x)) = x ∀x ∈ [0, 1]
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Examples
≥<
=ax
axxN
if0
if1 )(
1.0
x1.0a
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1.0
x1.0
==
=1 if0
0 if1 )(
x
xxN
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 Sugeno Negation
λ=20
λ=5
λ=0
λ=−0.9
N
x
)1(11
)( ∞−∈λλ+
−= ,x
xxN
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 Yager Negation
w=1
w=2 w=3
w=4
N
x
)0(1 )( ∞∈−= ,wxxN w w
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(t, s, N) system
x s y = N (N(x) t N(y))
x t y = N (N(x) s N(y))
∀x, y ∈ [0, 1]
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Examples
)1(1
1
)11(min
)1
10(max
∞−∈+−=
+−+=+
+−+=
,λλx
xN(x)
λxyy,xxsyλ
λxyyx,xty
xxN
yxxsy
yxxty
−===
1 )(
),max(
),min(
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