Chapter 2 The Operation of Fuzzy Setcomp.eng.ankara.edu.tr/files/2016/02/Chapter2.pdf · Fig 2.10 Disjunctive sum of two crisp sets Other operations in fuzzy set! Disjunctive sum

Chapter 2�The Operation of Fuzzy Set

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


!   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


!   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


!   Simple disjunctive sum(3)


!   (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


!   (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)}


!   Difference in fuzzy set n  Difference in crisp set

BABA ∩=−

A B

Fig 2.13 difference A – B


!   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


!   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


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


!   Hamming distance : distance and difference of 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-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 )

∪ •

+̂


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.


!   Duality of t-norms and t-conorms

Law sMorgane' Deby T

T

),(T1T

11

),(T 1),(

yxyx

yxyx

yxyx

yyxx

yxyx

⊥=

=⊥

−=

−=

−=

−=⊥

conormtyxnormtyx

−

−⊥

: T :

Chapter 2 The Operation of Fuzzy Setcomp.eng.ankara.edu.tr/files/2016/02/Chapter2.pdf · Fig 2.10 Disjunctive sum of two crisp sets Other operations in fuzzy set! Disjunctive sum

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