Chapter 2 The Operation of Fuzzy Set. 2.1 Standard operations of fuzzy set Complement set Union A B...

Chapter 2The Operation of Fuzzy Set

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

AAXAA

(1) Involution

(2) Commutativity A B = B AA 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 = AA A = A

(6) Absorption A (A B) = AA (A B) = A

(7) Absorption by X and A X = XA =

(8) Identity A = AA 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

a1

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

x1

1A

)(xA

x1

1A

)(xA

a1

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

0.00.1 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+cosa)

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

www aaC /1)1()(

),1( w

2.2 Fuzzy complement

2.2.3 Fuzzy Partition

(1)

(2)

(3)

),,,( 21 mAAA

iAi,

jiAA ji for ,

m

iA xXx

i1

1)(,

2.3 Fuzzy union

2.3.1 Axioms for union functionU : [0,1] [0,1] [0,1]

AB(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)

A1

X

B1

XAB

1

XFig 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 AB(x) = Max[A(x), B(x)]

),0( where])(,1[Min),( /1 wbabaU wwww

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 AB (Bold union)

Commutativity, associativity, identity, and De Morgan’s Law not idempotency, distributivity and absorption

2.3.3 Other union operations

BA )()()()()(, ˆ xxxxxXx BABABA

XXA

)]()(,1[Min)(, xxxXx BABA

XAAXXA ,

3) Drastic sum A B

4) Hamacher’s sum AB

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.

AB1

X

I[A(x), B(x)] = Min[A(x), B(x)], or

AB(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 wwww

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 .2 5 0 .5

1 0 0 .2 5 0 .5 I 2 (a ,b ) = 1 - M in [ 1 , 22 )1()1( ba ]

0 .7 5 0 0 .2 1 0 .4 4

0 .2 5 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)

xX, AB (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 AB

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) , )(x

B = 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 ASet BSet 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 ASet BSet 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 ASet BSet 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 = {(x△ 1, 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 x2x3 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

0.30.2

0.1

x1 x2x3 x4

Set ASet B

Bounded difference : shaded area

A

B0.4

Fig 2.15 bounded difference A B

2.5 Other operations in fuzzy set

AB(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

XxiiBiA

i

xx,1

)()(

A

x

1A(x)

B

x

1B(x)

B

A

x

1B(x)A(x)

B

A

x

1B(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

iBA xxBAe

1

2))()((),(

],1[,)()(),(/1

wxxBAdw

Xx

w

BAw

12.125.1015.00),( 2222 BAe

2.5.4 Cartesian product of fuzzy set

Power of fuzzy set

Cartesian product

Xxxx AA ,)]([)( 2

2

Xxxx mAAm ,)]([)(

)](,),([Min),,,( 121 121 nAAnAAA xxxxxnn

),(1xA ),(

2xA , )(x

nA 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-conormst-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 ( )

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.

2.6 t-norms and t-conorms

2.6.2 Duality of t-norms and t-conorms

Law sMorgane' Deby T

T

),(T1T

1

1

),(T 1),(

yxyx

yxyx

yxyx

yy

xx

yxyx

conormtyx

normtyx

: T

: