Chapter 2 The Operation of Fuzzy Setcomp.eng.ankara.edu.tr/files/2016/02/Chapter2.pdf · Fig 2.10...
Transcript of Chapter 2 The Operation of Fuzzy Setcomp.eng.ankara.edu.tr/files/2016/02/Chapter2.pdf · Fig 2.10...
Standard operations of Fuzzy Set
! Complement set
! Union A ∪ B
! Intersection A ∩ B
! difference between characteristics of crisp fuzzy set operator n law of contradiction n law of excluded middle
)(1)( xx AA µµ −=
)](),([Max)( xxx BABA µµµ =∪
)](),([Min)( xxx BABA µµµ =∩
A
∅=∩ AAXAA =∪
Fuzzy complement
! Requirements for complement function n Complement function
C: [0,1] → [0,1] (Axiom C1) C(0) = 1, C(1) = 0 (boundary condition) (Axiom C2) a,b ∈ [0,1]
if a < b, then C(a) ≥ C(b) (monotonic non-increasing) (Axiom C3) C is a continuous function. (Axiom C4) C is involutive.
C(C(a)) = a for all a ∈ [0,1]
))(()( xCx AA µµ =
Fuzzy complement
! Example of complement function
C(a) = 1 - a
a 1
C(a)
1
Fig 2.1 Standard complement set function
Fuzzy complement
! Example of complement function n standard complement set function
x 1
1 A
)(xAµ
x 1
1 A
)(xAµ
a 1
C(a)
1
t
⎩⎨⎧
>
≤=
tata
aCfor0for1
)(
Fuzzy complement
! Example of complement function(3)
It does not hold C3 and C4
Fuzzy union
! Axioms for union function U : [0,1] × [0,1] → [0,1] µA∪B(x) = U[µA(x), µB(x)]
(Axiom U1) U(0,0) = 0, U(0,1) = 1, U(1,0) = 1, U(1,1) = 1 (Axiom U2) U(a,b) = U(b,a) (Commutativity) (Axiom U3) If a ≤ a’ and b ≤ b’, U(a, b) ≤ U(a’, b’) Function U is a monotonic function. (Axiom U4) U(U(a, b), c) = U(a, U(b, c)) (Associativity) (Axiom U5) Function U is continuous. (Axiom U6) U(a, a) = a (idempotency)
A 1
X
B 1
X A∪B
1
X Fig 2.6 Visualization of standard union operation
Fuzzy union
! Examples of union function U[µA(x), µB(x)] = Max[µA(x), µB(x)], or µA∪B(x) = Max[µA(x), µB(x)]
1) Probabilistic sum (Algebraic sum)
n commutativity, associativity, identity and De Morgan’s law n
2) Bounded sum A⊕B (Bold union)
n Commutativity, associativity, identity, and De Morgan’s Law n n not idempotency, distributivity and absorption
Other union operations
BA +̂)()()()()(, ˆ xxxxxXx BABABA µµµµµ −+=∈∀ +
XXA =+̂
)]()(,1[Min)(, xxxXx BABA µµµ +=∈∀ ⊕
XAAXXA =⊕=⊕ ,
3) Drastic sum A B
4) Hamacher’s sum A∪B
∪•
⎪⎩
⎪⎨
⎧
=
=
=∈∀
othersfor,10)(when),(0)(when),(
)(, xxxx
xXx AB
BA
BA µµ
µµ
µ
0,)()()1(1
)()()2()()()(, ≥−−
−−+=∈∀ ∪ γ
µµγµµγµµ
µxx
xxxxxXxBA
BABABA
Other union operations
∪•
I:[0,1] × [0,1] → [0,1] )](),([)( xxIx BABA µµµ =∩
Fuzzy intersection
! Axioms for intersection function
(Axiom I1) I(1, 1) = 1, I(1, 0) = 0, I(0, 1) = 0, I(0, 0) = 0 (Axiom I2) I(a, b) = I(b, a), Commutativity holds. (Axiom I3) If a ≤ a’ and b ≤ b’, I(a, b) ≤ I(a’, b’),
Function I is a monotonic function. (Axiom I4) I(I(a, b), c) = I(a, I(b, c)), Associativity holds. (Axiom I5) I is a continuous function (Axiom I6) I(a, a) = a, I is idempotency.
A∩B 1
X
I[µA(x), µB(x)] = Min[µA(x), µB(x)], or µA∩B(x) = Min[µA(x), µB(x)]
Fuzzy intersection
! Examples of intersection n standard fuzzy intersection
1) Algebraic product (Probabilistic product)
∀x∈X, µA•B (x) = µA(x) • µB(x) n commutativity, associativity, identity and De Morgan’s law
2) Bounded product (Bold intersection)
n commutativity, associativity, identity, and De Morgan’s Law n n not idempotency, distributivity and absorption
Other intersection operations
BA•
∅=∅=∅ AAA ,
• BA]1)()(,0[Max)(, −+=∈∀ xxxXx BABA µµµ •
• •
3) Drastic product A B
4) Hamacher’s product A∩B
Other intersection operations
⎪⎩
⎪⎨
⎧
<
=
=
=
1)(),(when,01)(when),(1)(when),(
)(xx
xxxx
x
BA
BB
AA
BA
µµ
µµ
µµ
µ
0,))()()()()(1(
)()()( ≥−+−+
=∩ γµµµµγγ
µµµ
xxxxxxx
BABA
BABA
∩ •
∩ •
)()( BABABA ∩∪∩=⊕
A B
Fig 2.10 Disjunctive sum of two crisp sets
Other operations in fuzzy set
! Disjunctive sum
Other operations in fuzzy set
! Simple disjunctive sum )(xAµ = 1 - µA(x) , )(xBµ = 1 - µB(x)
),([)( xMinx ABA µµ =∩
)](1 xBµ−
)(1[)( xMinx ABA µµ −=∩ , )](xBµ
A ⊕ B = ),()( BABA ∩∪∩ then
),([{)( xMinMaxx ABA µµ =⊕)](1 xBµ− , )(1[ xMin Aµ− , )]}(xBµ
Other operations in fuzzy set
! Simple disjunctive sum(2) ex)
A = {(x1, 0.2), (x2, 0.7), (x3, 1), (x4, 0)}
B = {(x1, 0.5), (x2, 0.3), (x3, 1), (x4, 0.1)}
A = {(x1, 0.8), (x2, 0.3), (x3, 0), (x4, 1)}
B = {(x1, 0.5), (x2, 0.7), (x3, 0), (x4, 0.9)}
A ∩ B = {(x1, 0.2), (x2, 0.7), (x3, 0), (x4, 0)}
A ∩ B = {(x1, 0.5), (x2, 0.3), (x3, 0), (x4, 0.1)}
A ⊕ B = =∩∪∩ )()( BABA {(x1, 0.5), (x2, 0.7), (x3, 0), (x4, 0.1)}
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x2x1 x3 x4
0.2
0.5
0.3
0.7
1.0
0.1
0
Set ASet BSet A ⊕ B
Fig 2.11 Example of simple disjunctive sum
Other operations in fuzzy set
! Simple disjunctive sum(3)
Other operations in fuzzy set
! (Exclusive or) disjoint sum
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x2x1 x3 x4
0.2
0.5
0.3
0.7
1.0
0.1
0
Set ASet BSet A ⊕ B shaded area
Fig 2.12 Example of disjoint sum (exclusive OR sum)
)()()( xxx BABA µµµ −=Δ
Other operations in fuzzy set
! (Exclusive or) disjoint sum
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x2x1 x3 x4
0.2
0.5
0.3
0.7
1.0
0.1
0
Set ASet BSet A ⊕ B shaded area
Fig 2.12 Example of disjoint sum (exclusive OR sum)
)()()( xxx BABA µµµ −=Δ
A = {(x1, 0.2), (x2, 0.7), (x3, 1), (x4, 0)} B = {(x1, 0.5), (x2, 0.3), (x3, 1), (x4, 0.1)} A△B = {(x1, 0.3), (x2, 0.4), (x3, 0), (x4, 0.1)}
Other operations in fuzzy set
! Difference in fuzzy set n Difference in crisp set
BABA ∩=−
A B
Fig 2.13 difference A – B
Other operations in fuzzy set
! Simple difference
ex)
)](1),([ xxMin BABABA µµµµ −== ∩−
A = {(x1, 0.2), (x2, 0.7), (x3, 1), (x4, 0)}
B = {(x1, 0.5), (x2, 0.3), (x3, 1), (x4, 0.1)}
B = {(x1, 0.5), (x2, 0.7), (x3, 0), (x4, 0.9)}
A – B = A ∩ B = {(x1, 0.2), (x2, 0.7), (x3, 0), (x4, 0)}
A
B
0.7
0.2
1
0.7
0.5
0.3
0.2
0.1
x1 x2 x3 x4
Set A
Set B
Simple difference A-B : shaded area
Fig 2.14 simple difference A – B
Other operations in fuzzy set
! Simple difference(2)
1
0.7
0.5
0.3
0.2
0.1
x1 x2 x3 x4
Set A
Set B
Bounded difference : shaded area
A
B 0.4
Fig 2.15 bounded difference A θ B
Other operations in fuzzy set
µAθB(x) = Max[0, µA(x) - µB(x)] ! Bounded difference
A θ B = {(x1, 0), (x2, 0.4), (x3, 0), (x4, 0)}
Distance in fuzzy set
! Hamming distance d(A, B) =
1. d(A, B) ≥ 0 2. d(A, B) = d(B, A) 3. d(A, C) ≤ d(A, B) + d(B, C) 4. d(A, A) = 0 ex) A = {(x1, 0.4), (x2, 0.8), (x3, 1), (x4, 0)}
B = {(x1, 0.4), (x2, 0.3), (x3, 0), (x4, 0)} d(A, B) = |0| + |0.5| + |1| + |0| = 1.5
∑∈=
−n
XxiiBiA
i
xx,1
)()( µµ
A
x
1 µA(x)
B
x
1 µB(x)
B
A
x
1 µB(x) µA(x)
B
A
x
1 µB(x) µA(x)
distance between A, B difference A- B
Distance in fuzzy set
! Hamming distance : distance and difference of fuzzy set
Distance in fuzzy set
! Euclidean distance ex)
! Minkowski distance
∑=
−=n
iBA xxBAe
1
2))()((),( µµ
],1[,)()(),(/1
∞∈⎟⎠
⎞⎜⎝
⎛−= ∑
∈
wxxBAdw
Xx
wBAw µµ
12.125.1015.00),( 2222 ==+++=BAe
Cartesian product of fuzzy set
! Power of fuzzy set
! Cartesian product
Xxxx AA ∈∀= ,)]([)( 22 µµ
Xxxx mAAm ∈∀= ,)]([)( µµ
)](,),([Min),,,( 121 121 nAAnAAA xxxxxnn
µµµ ……… =×××
),(1
xAµ ),(2
xAµ ,… )(xnAµ as membership functions of A1, A2,…, An
for ,11 Ax ∈∀ ,22 Ax ∈ nn Ax ∈,… .
t-norms and t-conorms
Definitions for t-norms and t-conorms ! t-norm
T : [0,1]×[0,1]→[0,1] ∀x, y, x’, y’, z ∈ [0,1]
i) T(x, 0) = 0, T(x, 1) = x : boundary condition ii) T(x, y) = T(y, x) : commutativity iii) (x ≤ x’, y ≤ y’) → T(x, y) ≤ T(x’, y’) : monotonicity iv) T(T(x, y), z) = T(x, T(y, z)) : associativity
1) intersection operator ( ∩ ) 2) algebraic product operator ( • ) 3) bounded product operator ( ) 4) drastic product operator ( )
•
∩ •
t-norms and t-conorms
! t-conorm (s-norm) T : [0,1]×[0,1]→[0,1] ∀x, y, x’, y’, z ∈ [0,1]
i) T(x, 0) = 0, T(x, 1) = 1 : boundary condition ii) T(x, y) = T(y, x) : commutativity iii) (x ≤ x’, y ≤ y’) → T(x, y) ≤ T(x’, y’) : monotonicity
iv) T(T(x, y), z) = T(x, T(y, z)) : associativity 1) union operator ( ∪ ) 2) algebraic sum operator ( ) 3) bounded sum operator ( ⊕ ) 4) drastic sum operator ( ) 5) disjoint sum operator ( Δ )
∪ •
+̂
t-norms and t-conorms
Ex) a) ∧ : minimum
Instead of *, if ∧ is applied x ∧ 1 = x Since this operator meets the previous conditions, it is a t-norm.
b) ∨ : maximum
If ∨ is applied instead of *, x ∨ 0 = x then this becomes a t-conorm.