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2.1 Standard operations of fuzzy set
Complement set
Union A B
Intersection A B
difference between characteristics of crisp fuzzy set operator
law of contradiction
law of excluded middle
)(1)( xx AA
)](),([Max)( xxx BABA
)](),([Min)( xxx BABA
A
AA
XAA
AA
BABA BABA
)()()()( BABABABA
)()()()( BABABABA
(1) Involution
(2) Commutativity
A B = B A
A B = B A
(3) Associativity
(A B) C = A (B C) (A B) C = A (B C) (4) Distributivity
A (B C) = (A B) (A C) A (B C) = (A B) (A C)
(5) Idempotency
A A = A
A A = A
(6) Absorption
A (A B) = A
A (A B) = A
(7) Absorption by X and
A X = X
A =
(8) Identity
A = A
A X = A
(9) De Morgan’s law
(10) Equivalence formula
(11) Symmetrical difference formula
AA
BABA BABA
)()()()( BABABABA
)()()()( BABABABA
Table 2.1 Characteristics of standard fuzzy set operators
2.2 Fuzzy complement
2.2.1 Requirements for complement function
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
2.2 Fuzzy complement
2.2.2 Example of complement function(1)
C(a) = 1 - a
a 1
C(a)
1
Fig 2.1 Standard complement set function
2.2 Fuzzy complement
2.2.2 Example of complement function(2)
standard complement set function
x 1
1 A
)(xA
x 1
1 A
)(xA
a 1
C(a)
1
t
ta
taaC
for0
for1)(
2.2 Fuzzy complement
2.2.2 Example of complement function(3)
It does not hold C3 and C4
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
C(a)
2.2 Fuzzy complement
2.2.2 Example of complement function(4) Continuous fuzzy complement function C(a) = 1/2(1+cos a)
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Cw(a)
w=0.5
w=1
w=2
w=5
a
2.2 Fuzzy complement
2.2.2 Example of complement function(5)
Yager complement function
ww
w aaC /1)1()(
),1(w
2.2 Fuzzy complement
2.2.3 Fuzzy Partition
(1)
(2)
(3)
),,,( 21 mAAA
iAi,
jiAA ji for ,m
i
A xXxi
1
1)(,
2.3 Fuzzy union
2.3.1 Axioms for union function
U : [0,1] [0,1] [0,1]
A B(x) = U[ A(x), B(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
2.3 Fuzzy union
2.3.2 Examples of union function
U[ A(x), B(x)] = Max[ A(x), B(x)], or A B(x) = Max[ A(x), B(x)]
),0( where])(,1[Min),( /1 wbabaU www
w
2.3 Fuzzy union
Yager’s union function :holds all axioms except U6.
0 0.25 0.5
1 1 1 1 U1(a,b) = Min[1, a+b]
0.75 0.75 1 1
0.25 0.25 0.5 0.75 w = 1
a
0 0.25 0.5
1 1 1 1 U2(a,b) = Min[1,22 ba ]
0.75 0.75 0.79 0.9
0.25 0.25 0.35 0.55 w = 2
a
0 0.25 0.5
1 1 1 1 U (a,b) = Max[ a, b] : standard union function
0.75 0.75 0.75 0.75
0.25 0.25 0.25 0.5 w
a
1) Probabilistic sum (Algebraic sum)
commutativity, associativity, identity and De Morgan’s law
2) Bounded sum A B (Bold union)
Commutativity, associativity, identity, and De Morgan’s Law
not idempotency, distributivity and absorption
2.3.3 Other union operations
BA ˆ
)()()()()(, ˆ xxxxxXx BABABA
XXA ˆ
)]()(,1[Min)(, xxxXx BABA
XAAXXA ,
3) Drastic sum A B
4) Hamacher’s sum A B
othersfor,1
0)(when),(
0)(when),(
)(, xx
xx
xXx AB
BA
BA
0,)()()1(1
)()()2()()()(,
xx
xxxxxXx
BA
BABABA
2.3.3 Other union operations
I:[0,1] [0,1] [0,1]
)](),([)( xxIx BABA
2.4 Fuzzy intersection
2.4.1 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[ A(x), B(x)] = Min[ A(x), B(x)], or
A B(x) = Min[ A(x), B(x)]
2.4 Fuzzy intersection
2.4.2 Examples of intersection
standard fuzzy intersection
),0(],))1()1((,1[Min1),( /1 wbabaI www
w
2.4 Fuzzy intersection
Yager intersection function
B 0 0.25 0.5
1 0 0.25 0.5 I1(a,b) =1-Min[1, 2-a-b]
0.75 0 0 0.25
0.25 0 0 0 w = 1
a
B
0 0.25 0.5
1 0 0.25 0.5 I2(a,b) = 1-Min[1, 22 )1()1( ba ]
0.75 0 0.21 0.44
0.25 0 0 0.1 w = 2
a
0 0.25 0.5
1 0 0.25 0.5 I (a,b) = Min[ a, b]
0.75 0 0.25 0.5
0.25 0 0.25 0.25 w
a
1) Algebraic product (Probabilistic product)
x X, A B (x) = A(x) B(x)
commutativity, associativity, identity and De Morgan’s law
2) Bounded product (Bold intersection)
commutativity, associativity, identity, and De Morgan’s Law
not idempotency, distributivity and absorption
2.4.3 Other intersection operations
BA
AAA ,
BA
]1)()(,0[Max)(, xxxXx BABA
3) Drastic product A B
4) Hamacher’s product A B
2.4.3 Other intersection operations
1)(),(when,0
1)(when),(
1)(when),(
)(
xx
xx
xx
x
BA
BB
AA
BA
0,))()()()()(1(
)()()(
xxxx
xxx
BABA
BABA
)()( BABABA
A B
Fig 2.10 Disjunctive sum of two crisp sets
2.5 Other operations in fuzzy set
2.5.1 Disjunctive sum
2.5 Other operations in fuzzy set
Simple disjunctive sum
)(xA
= 1 - A(x) , )(xB
= 1 - B(x)
),([)( xMinx ABA)](1 xB
)(1[)( xMinx ABA , )](xB
A B = ),()( BABA then
),([{)( xMinMaxx ABA )](1 xB , )(1[ xMin A , )]}(xB
2.5 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 A
Set B
Set A B
Fig 2.11 Example of simple disjunctive sum
2.5 Other operations in fuzzy set
Simple disjunctive sum(3)
2.5 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 A
Set B
Set A B shaded area
Fig 2.12 Example of disjoint sum (exclusive OR sum)
)()()( xxx BABA
2.5 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 A
Set B
Set A B shaded area
Fig 2.12 Example of disjoint sum (exclusive OR sum)
)()()( xxx BABA
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)}
2.5 Other operations in fuzzy set
2.5.2 Difference in fuzzy set
Difference in crisp set
BABA
A B
Fig 2.13 difference A – B
2.5 Other operations in fuzzy set
Simple difference
ex)
)](1),([ xxMin BABABA
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
2.5 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 B
2.5 Other operations in fuzzy set
A B(x) = Max[0, A(x) - B(x)] Bounded difference
A B = {(x1, 0), (x2, 0.4), (x3, 0), (x4, 0)}
2.5.3 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
Xxi
iBiA
i
xx,1
)()(
A
x
1 A(x)
B
x
1 B(x)
B
A
x
1 B(x) A(x)
B
A
x
1 B(x) A(x)
distance between A, B difference A- B
2.5.3 Distance in fuzzy set
Hamming distance : distance and difference of fuzzy set
2.5.3 Distance in fuzzy set
Euclidean distance
ex)
Minkowski distance
n
i
BA xxBAe1
2))()((),(
],1[,)()(),(
/1
wxxBAd
w
Xx
w
BAw
12.125.1015.00),( 2222BAe
2.5.4 Cartesian product of fuzzy set
Power of fuzzy set
Cartesian product
Xxxx AA,)]([)( 2
2
Xxxx m
AAm ,)]([)(
)](,),([Min),,,( 121 121 nAAnAAA xxxxxnn
),(1
xA ),(2
xA , )(xnA as membership functions of A1, A2, , An
for ,11 Ax ,22 Ax nn Ax, .
2.6 t-norms and t-conorms
2.6.1 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 ( )
2.6 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 ( )
ˆ
2.6 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.