lectures 2&3: fuzzy sets

In the Name of God

Lectures 2&3: Fuzzy Sets

Fuzzy Logicy gWhat is Fuzzy Logic (FL)?

It is able to simultaneously handle numerical data and linguistic

FL is in fact, a precise problema precise problem--solving methodologysolving methodology.

knowledge.

A technique that facilitates the control of a complicated system without knowledge of its mathematical description

Fuzzy logic differs from classical logic in that statements are no longer black or white, true or false, on or off.

knowledge of its mathematical description.

, ,In traditional logic, an object takes on a value of either zero or one. In fuzzy logic, a statement can assume any real value between 0 and 1, representing the degree to which an element belongs to a given set.

Fuzzy Logicy g

History of Fuzzy Logic

Professor Lotfi A. Zadehhttp://www.cs.berkeley.edu/~zadeh/p // y / /

In 1965, Lotfi A. Zadeh of the University of California at Berkeley published "Fuzzy Sets," which laid out the mathematics of fuzzy set theory and, by extension, fuzzy logic. Zadeh had observed that conventional computer logic could not manipulate data that represented subjective or vague ideas so he created fuzzy logic to allowdata that represented subjective or vague ideas, so he created fuzzy logic to allow computers to determine the distinctions among data with shades of gray, similar to the process of human reasoning.

Pioneering worksg

20 years after its conception• Interest in fuzzy systems was sparked by Seiji Yasunobu and Soji

Miyamoto of Hitachi, who in 1985 provided simulations that demonstrated the superiority of fuzzy control systems for the Sendai railway Their ideas were adopted and fuzzy systems wereSendai railway. Their ideas were adopted, and fuzzy systems were used to control accelerating and braking when the line opened in 1987.

• Also in 1987, during an international meeting of fuzzy researchers in Tokyo Takeshi Yamakawa demonstrated the use of fuzzy controlTokyo, Takeshi Yamakawa demonstrated the use of fuzzy control, through a set of simple dedicated fuzzy logic chips, in an "inverted pendulum" experiment. This is a classic control problem, in which a vehicle tries to keep a pole mounted on its top by a hinge upright by moving back and forthmoving back and forth.

• Observers were impressed with this demonstration, as well as later experiments by Yamakawa in which he mounted a glass containing water to the top of the pendulum. The system maintained stability in b th Y k t ll t t i hiboth cases. Yamakawa eventually went on to organize his own fuzzy-systems research lab to help exploit his patents in the field.

http://en.wikipedia.org/wiki/Fuzzy_control_system

Uncertaintyy

▫ Probability theory is capable of representing only one of several distinct types of uncertainty.

▫ When A is a fuzzy set and x is a relevant object, the proposition “x is a member of A” is not necessarilyproposition x is a member of A is not necessarily either true or false. It may be true only to some degree, the degree to which x is actually a member of A.

▫ For example: the weather today Sunny: If we define any cloud cover of 25% or less is sunny. This means that a cloud cover of 26% is not sunny? “Vagueness” should be introduced.

▫ The possibility theory

Classical Sets

• Classical sets have crisp boundaries

• Example: Numbers greater than 6

• Clear, Unambiguous boundary

The crisp sets vs the fuzzy setsy

• The crisp set is defined in such a way as to dichotomize the individuals in some given universe of discourse into two groups: members and nonmembers.▫ However, many classification concepts do not exhibit this y p

characteristic.▫ For example, the set of tall people, expensive cars, or

sunny days.sunny days.• A fuzzy set can be defined mathematically by assigning to

each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy setrepresenting its grade of membership in the fuzzy set.▫ For example: a fuzzy set representing our concept of sunny

might assign a degree of membership of 1 to a cloud cover of 0% 0 8 to a cloud cover of 20% 0 4 to a cloud cover ofof 0%, 0.8 to a cloud cover of 20%, 0.4 to a cloud cover of 30%, and 0 to a cloud cover of 75%.

Classical (Crisp) Sets( )

• Classical sets are suitable for various applications

• Classical sets have proven to be an important t l f th ti d t itool for mathematics and computer science

• They do NOT reflect human concepts and thoughtsthoughts

• Human thought tend to be abstract and impreciseimprecise

Crisp Sets

• Example: using crisp sets for “tall person” set

• Example:

It i t l d i d t• It is unnatural and inadequate• 180.0001 is tall but 179.999 is Not!

Thi di ti ti i bl• This distinction is unreasonable.• Classical Set’s flaw: Sharp transition between

inclusion and exclusion setinclusion and exclusion set.

Crisp sets: an overview

• Three basic methods to define sets:


▫ The list method: a set is defined by naming all its members.

▫ The rule method: a set is defined by a property satisfied by its},...,,{ 21 naaaA

The rule method: a set is defined by a property satisfied by its members.

where ‘|’ denotes the phrase “such that”)}(|{ xPxA

where | denotes the phrase such thatP(x): a proposition of the form “x has the property P ”

▫ A set is defined by a characteristic function. Axfor1

the characteristic function

AxAx

xA for 0for 1

)(

}10{: XA }1,0{: XA

Crisp sets: an overviewCrisp sets: an overview

• A family of sets: a set whose elements are sets▫ It can be defined in the form:

where i and I are called the set index and the index set, respectively.}|{ IiAi

▫ The family of sets is also called an indexed set.▫ For example: A

},...,,{ 21 nAAAA

• A is a subset of B:• A and B are equal sets:

}, ,,{ 21 n

BABA ABBA and

• A and B are not equal:• A is proper subset of B:• A is included in B:

BA

BABABABA and


• The power set of A, ( ): the family of all subsets of a given set A.)(AP


▫ The second order power set of A: ▫ The higher order power set of A:

• The cardinality of A (|A|): the number of members of a finite set A.

))(()( AA PPP2 ),...(),( 43 AA PP

▫ For example:

• B – A: the relative complement of a set A with respect to set B

,2|)(| ||AA P| |22|)(|

A

A 2P

▫ If the set B is the universal set, then▫

},|{ AxBxxAB .AAB

AA ▫▫

AAXX


• The union of sets A and B:


• The generalized union operation: for a family of sets,}or |{ BxAxxBA

}somefor|{ IiAxxA

• The intersection of sets A and B:

}somefor |{ IiAxxA iiIi

}d|{ BABA• The generalized intersection operation: for a family of sets,

}and|{ BxAxxBA

} allfor |{ IiAxxA iiIi

• Disjoint: any two sets that have no common members

Ii

BA BA

Supremum and infimum• Let R denote a set of real number.▫ If there is a real number r such that for every thenRrx

Supremum and infimum

▫ If there is a real number r such that for every , then r is called an upper bound of R, and A is bounded above by r.

▫ If there is a real number s such that for every , then s is called an lower bound of R, and A is bounded below by s.

Rxrx

sx Rx, y

• For any set of real numbers R that is bounded above, a real number r is called the supremum of R (write r = sup R) iff:(a) r is an upper bound of R;(b) no number less than r is an upper bound of R (R is the

smallest upper bound).For an set of real n mbers R that is bo nded belo a real• For any set of real numbers R that is bounded below, a real number s is called the infimum of R (write s = inf R) iff:(a) s is an lower bound of R;(b) no number greater than s is an lower bound of R (R is the(b) no number greater than s is an lower bound of R (R is the

largest lower bound).

Fuzzy Setsy

• Sets with fuzzy boundaries• Sets with fuzzy boundaries

A = Set of tall people

1.0

Crisp set A Fuzzy set A1.0

Membershipfunction

.5

.9

Heights180 cm Heights180 190

Fuzzy Setsy• A membership function: ▫ A characteristic function: the values assigned to the elements of▫ A characteristic function: the values assigned to the elements of

the universal set fall within a specified range and indicate the membership grade of their elements in the set.

▫ Larger values denote higher degrees of set membership▫ Larger values denote higher degrees of set membership.• A set defined by membership functions is a fuzzy set.• The most commonly used range of values of membership

functions is the unit interval [0 1]functions is the unit interval [0,1].• Notation:

▫ The membership function of a fuzzy set A is denoted by :A]10[: X

▫ In the other one, the function is denoted by A and has the same form

▫ In this course we use the second notation]1,0[: XA

]1,0[: XA

▫ In this course, we use the second notation.

Membership Functions (MFs)( )

• Characteristics of MFs:▫ Subjective measures▫ Not probability functions

MFs “tall” in China

5

.8“tall” in the Norway

Heights178 cm

.5

.1

y

“tall” in NBA

Heights178 cm

Fuzzy Setsy

• Formal definition:• Formal definition:A fuzzy set A in X is expressed as a set of ordered

pairs:

A x x x XA {( , ( ))| }

X: Universe oruniverse of discourse

Fuzzy setMembership

function(MF)

A fuzzy set is totally characterized by aA fuzzy set is totally characterized by amembership function (MF).membership function (MF).

Fuzzy Sets with Discrete Universes

• Fuzzy set C = “desirable city to live in”X {T h T b i R ht} (di t d d d)X = {Tehran, Tabriz, Rasht} (discrete and non-ordered)C = {(Tehran, 0.1), (Tabriz, 0.8), (Rasht, 0.9)}

• Fuzzy set A = “sensible number of children in a f il ”family”X = {0, 1, 2, 3, 4, 5, 6} (discrete ordered universe)A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}

Fuzzy Sets with Cont. Universesu y Se s Co U e ses

• Fuzzy set B = “about 50 years old”• Fuzzy set B = about 50 years oldX = Set of positive real numbers (continuous)B = {(x, B(x)) | x in X}

B xx

( )

1

1 5010

2

10

Alternative Notation

• A fuzzy set A can be alternatively denoted as• A fuzzy set A can be alternatively denoted as follows:

A x x ( ) /A x xAx X

i ii

( ) /

A x x ( ) /

X is discrete

A x xAX

( ) /X is continuous

Note that and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.does not imply division.

Alternative Notation

• Examples:Examples:

A is discrete

C is discrete

B is continuous

Fuzzy Partitiony

• Fuzzy partitions formed by the linguistic values y p y g“young”, “middle aged”, and “old”:

Fuzzy Partitiony

• Fuzzy partitions formed by the linguistic values y p y g“very low”, “low”, “medium”, “high”, and “very high”:

Some Definitions

• Linguistic variable • Convexity• Linguistic variable• Linguistic value• Support

• Convexity• Fuzzy numbers• Bandwidthpp

• Core• Normality

• Symmetricity• Open left or right, closed

• Crossover points• Fuzzy singleton

cut strong cut• -cut, strong -cut

Linguistic termsg

• Linguistic variable• Linguistic variable▫ In the first example “age” and in the second one

“temperature” are linguistic variables.

• Linguistic valueI th fi t l “ ” d i th d▫ In the first example “young” and in the second one “high” are linguistic values.

Fuzzy Variable y

• A fuzzy variable is defined by the quadrupley y q pV = { x, l, u, m}

• X is the variable symbolic name: temperature• L is the set of labels: low, medium and high• U is the universe of discourseuniverse of discourse• M are the semantic rules that define the

meaning of each label in L (membership f ti )functions).

Fuzzy Variable Exampley

• X = Temperaturep• L = {low, medium, high}• U = {xX | -70o <= x <= +70o}• M =

1.0low medium high

0.0

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70

Membership Functions?

• Subjective evaluation: The shape of the j pfunctions is defined by specialists

• Ad-hoc: choose a simple function that is suitable to solve the problem

• Distributions, probabilities: information extracted f tfrom measurements

• Adaptation: testingAutomatic: algorithms used to define functions• Automatic: algorithms used to define functions from data

Variable Terminology gy

• Completude: A variable is complete if for any x p p yX there is a fuzzy set such as μ(x) > 0

1.0

0 0

Complete

0.0

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70

1.0

0.0

Incomplete

-70 -60 -50 -40 -30 -20 -10 0 10 20 30 40 50 60 70

Partition of Unityy

• A fuzzy variable forms a partition of unity if for each input l

pvalue x

• where p is the number of sets to which x belongs

p

iA x

i1

1)(

p g• There is no rule to define the overlapping degree

between two neighbouring sets• A rule of thumb is to use 25% to 50%A rule of thumb is to use 25% to 50%

1,01,0

0,0

0,5

0,0

0,5

-70 -60-50-40-30-20-10 0 10 20 30 40 50 60 70No Partition of

Unity

-70 -60-50 -40-30-20-10 0 10 20 30 40 50 60 70Partition of

Unity

MF Terminologygy

MF

1

.5

X0

Core

Crossover points

Crossover points

Support

- cut

Support

MF Terminologygy

S t MF

1

• Support• The Support of a fuzzy

set A is the set of all.5

0

set A is the set of all points x in X such that:

X0 Core

Crossover points

- cutI h d

Support• In other words:

MF Terminologygy

C MF

1

• Core• The Core of a fuzzy set

A is the set of all points.5

0

A is the set of all points x in X such that:

X0 Core

Crossover points

- cutI h d


MF Terminologygy

C P i t MF

1

• Crossover Point• The Crossover point of

a fuzzy set A is a point.5

0

a fuzzy set A is a point x in X such that:

X0 Core

Crossover points

- cutI h d


MF Terminologygy

MF

1

•• is a crisp set defined by:

.5

0

X0 Core

Crossover points

- cut

•

Support

••

Indeed …

• The α-cut of a fuzzy set A is the crisp set that contains all th l t f th i l t X h b hithe elements of the universal set X whose membership grades in A are greater than or equal to the specified value of α

• The strong α -cut of a fuzzy set A is the crisp set that contains all the elements of the universal set X whose membership grades in A are only greater than themembership grades in A are only greater than the specified value of α

MF Terminologygy

MF

1• Normality:• Fuzzy set A is normal if

.5

0a

• Fuzzy set A is normal if its core is non-empty.

X0 Core

Crossover points

a - cut

• In other words:• We can always find a

i i X f hi hSupport

point x in X for which:

MF TerminologygyMF

1

.5

1

• Fuzzy Singleton:• A fuzzy set whose

X0 Core

Crossover points

• A fuzzy set whose support is a single point in X is a fuzzy singleton

Support

- cutif:

• Example:• A fuzzy singleton• A fuzzy singleton• “45 years old”

MF Terminologygy

• The height of a fuzzy set A:g y▫ The height of a fuzzy set A is the largest membership grade

obtained by any element in that set.

)()( AAh

▫ A fuzzy set A is called normal when h(A) = 1.It i ll d b l h h(A) <1

)(sup)( xAAhXx

▫ It is called subnormal when h(A) <1.▫ The height of A may also be viewed as the supremum of α

for which Aα ≠ Φ.

MF Terminologygy

• Scalar cardinality: The cardinality of a fuzzy set is equal t th f th b hi d f ll l tto the sum of the membership degrees of all elements.

• The cardinality is represented by |A|

n

• For example, the scalar cardinality of fuzzy set D2 is

i

iA xA1

)(||

p y y 2|D2| = 2(0.13+0.27+0.4+0.53+0.67+0.8+0.93)+5 = 12.46

Distance

• The distance dp between two sets represented by points in the space is defined aspoints in the space is defined as

pn

i

piBiA

p xxBAd 1

|)()(|),(

• If p=2, the distance is the Euclidean distance, if p=1 the distance it is the Hamming distance

i1

g• If the point B is the empty set (the origin)

1 ( , ) ( ) 0n

A id A x 1

1

1

( , ) | | ( )

in

A ii

d A A x

• So, the cardinality of a fuzzy set is the Hamming distance to the origin

Convex function

• In mathematics, a real-valued ,function f defined on an interval is called convex, if for any two points x and y in its domain Cpoints x and y in its domain Cand any t in [0,1], we have

• In other words a function is• In other words, a function is convex if and only if its epigraph (the set of points lying on or above the graph) is a convex set.

Concave function

• In mathematics, a concave function is the negative of , ga convex function.

• Formally, a real-valued function f defined on an interval is called concave if for any two points x and y in itsis called concave, if for any two points x and y in its domain C and any t in [0,1], we have

Convexity of Fuzzy Setsy y

• A fuzzy set A is convex if for any in [0, 1],A fuzzy set A is convex if for any in [0, 1], A A Ax x x x( ( ) ) min( ( ), ( ))1 2 1 21

Alternatively, A is convex is all its a-cuts are convex.

MF Terminologygy

F b• Fuzzy number:• A fuzzy number A is a fuzzy set that satisfies the

condition for normality and convexitycondition for normality and convexity.• Most fuzzy sets in the literature satisfy these

conditions, and thus most fuzzy sets are fuzzy numbers.

S t• Symmetry:• A fuzzy set A is symmetric if

its MF is symmetric aroundits MF is symmetric around a certain point x=c,

MF Terminologygy

B d idth f l d f t• Bandwidth of normal and convex fuzzy sets:• The bandwidth or simply width of a fuzzy set is

defined as the distance between its unique (whydefined as the distance between its unique (why unique?) crossover points

• where

• Open left – Open right:• A fuzzy number A is open left if :• A fuzzy number A is open left if :

Crisp sets …

Set-Theoretic Operations

• Containment or Subset:A B A B

• Sets contain subsets• Sets contain subsets.• A is a subset of B (AB) iff

every element of A is anevery element of A is an element of B.

• A is a subset of B iff A belongs to the power set of BA is a subset of B iff there is• A is a subset of B iff there is no element of A that does not belong to B


• Complement:pA X A x x

A A ( ) ( )1


• Union:C A B x x x x xc A B A B ( ) max( ( ), ( )) ( ) ( )


• Intersection:

C A B x x x x xc A B A B ( ) min( ( ), ( )) ( ) ( )

More example▫ A1, A2, A3 are normal.▫ B and C are subnormal

21 AAB Normality and convexity▫ B and C are subnormal.

▫ B and C are convex.▫ are not

convexCBCB and

Normality and convexitymay be lost when we operate on fuzzy sets by the standard operations of intersection andconvex.

32 AAC of intersection and complement.


• Cartesian productp

( , ) min( ( ), ( ))A B A Bx y x y

• Cartesian co-product

( ) max( ( ) ( ))x y x y ( , ) max( ( ), ( ))A B A Bx y x y

Note that these two operations are defined for two-dimensional MFs

MF Formulation (one dimensional)( )

0minmax);( xcaxcbaxtrimfTriangular MF:

0,,minmax),,;(bcab

cbaxtrimf

T id l

01minmax);( xdaxdcbaxtrapmf

Triangular MF:

Trapezoidal MF:

0,,1,minmax),,,;(cdab

dcbaxtrapmf

21

cx

1

Gaussian MF: 2),;(

aecaxgaussmf

Generalized bell MF:

b

acx

cbaxgbellmf 2

1

1),,;(

MF Formulation (one dimensional)

• Generalized Bell MF:

( )

bcxcbaxgbellmf 2

1),,;(

• Specified by three parameters: {a, b, c}a

cx1


• Consider three fuzzy sets that represent the concepts of a young, middle-aged, and old person. The membership functions are defined on the interval [0,80] as follows:


• Sigmoidal MF: )(

1),;( caxsigmf Sigmoidal MF: )(1),;( cxae

g f

Extensions:

Absolute difference of two sig MFsof two sig. MFs

Product of two sig. MFs


cxxcF L ,

cxcxF

cxFcxLR

R

L

,

,),,;(

Left-right (L-R) MF:

Example: )1,0max()( 2xxFL )exp()( 3xxFR

c=65 c=25a=60b=10

a=10b=40

MF Formulation (two dimensional)( )

Cylindrical Extension of one-dimensional MFsy

If A is a fuzzy set in X, then its Cylindrical Extension c(A) is a fuzzy set in X × Y defined as

l tcyl_ext.m

2D MF Projectionj

Two-dimensional Projection ProjectionTwo dimensionalMF

Projectiononto X

Projectiononto Y

R x y( , ) A x( ) B y( ) R

y R x ymax ( , )

x R x ymax ( , )

2D MFs

Derivatives of parameterized MFs

• In some cases, the fuzzy system needs to be d tiadaptive

• We need the derivative of the MFs with respect to inputs (linguistic variables) andrespect to inputs (linguistic variables) and parameters

• It has central role in the learning (we needIt has central role in the learning (we need some sort of learning) in adaptive systems


• For the Gaussian MFs as

Th d i ti• The derivatives are


• For the bell MFs as

Th d i ti• The derivatives are

More on fuzzy set operationsy

• Three basic operations▫ Fuzzy complement▫ Fuzzy union▫ Fuzzy intersection▫ Fuzzy intersection

• We mentioned basic forms of these operationsp

• However, more general forms can be considered

Fuzzy Complementy

• General requirements:• General requirements:▫ Boundary: c(0) = 1 and c(1) = 0▫ Monotonicity: c(a) ≥ c(b) if a ≤ b

I l ti ( ( ))▫ Involution: c(c(a)) = a• Two types of fuzzy complements:▫ Sugeno’s complement (s > -1):g p ( )

Yager’s complement (w > 0):

1( )1s

ac asa

▫ Yager s complement (w > 0):

1/( ) (1 )w wwc a a

Fuzzy Complementy

Sugeno’s complement: Yager’s complement:1( )1s

ac asa

1/( ) (1 )w w

wc a a

Fuzzy Complementy

• Several important properties are shared by all fuzzy l tcomplements.

▫ Equilibrium of a fuzzy complement c: any value a for which c(a) = a.

Fuzzy Complementy

• Dual point:Dual point:▫ If we are given a fuzzy complement c and a membership grade

whose value is represented by a real number then any membership grade represented by the real number

],1,0[a]1,0[admembership grade represented by the real number

such that]1,0[a

)()( acaaac dd

is called a dual point of a with respect to c. ▫ There is at most one dual point for each particular fuzzy

complement c and membership grade of value acomplement c and membership grade of value a.

Fuzzy Intersection: T-normy

• Basic requirements:q▫ Boundary: T(0, 0) = 0, T(a, 1) = T(1, a) = a▫ Monotonicity: T(a, b) ≤ T(c, d) if a ≤ c and b ≤ d▫ Commutativity: T(a, b) = T(b, a)▫ Associativity: T(a, T(b, c)) = T(T(a, b), c)F l• Four examples:▫ Minimum: Tm(a, b)▫ Algebraic product: T (a b)▫ Algebraic product: Ta(a, b)▫ Bounded product: Tb(a, b)▫ Drastic product: Td(a, b)p ( , )

Fuzzy Intersection: T-normy

T-norm Operator

Algebraic Bounded DrasticMinimum:

Tm(a, b)

gproduct:Ta(a, b)

product:Tb(a, b)

product:Td(a, b)

Fuzzy Union: T-conorm or S-norm

• Basic requirements:q▫ Boundary: S(1, 1) = 1, S(a, 0) = S(0, a) = a▫ Monotonicity: S(a, b) ≤ S(c, d) if a ≤ c and b ≤ d▫ Commutativity: S(a, b) = S(b, a)▫ Associativity: S(a, S(b, c)) = S(S(a, b), c)F l• Four examples▫ Maximum: Sm(a, b)▫ Algebraic sum: S (a b)▫ Algebraic sum: Sa(a, b)▫ Bounded sum: Sb(a, b)▫ Drastic sum: Sd(a, b)( , )

Fuzzy Union: T-conorm or S-norm

T-conorm or S-norm

Algebraic Bounded DrasticMaximum:

Sm(a, b)

gsum:

Sa(a, b)sum:

Sb(a, b)sum:

Sd(a, b)

Generalized DeMorgan’s Lawg

• T-norms and T-conorms are duals which supports the generalization of DeMorgan’s law:▫ T(a, b) = c(S(c(a), c(b)))▫ S(a, b) = c(T(c(a), c(b)))

Tm(a, b)Ta(a, b)Tb(a, b)

Sm(a, b)Sa(a, b)Sb(a, b)Tb(a, b)

Td(a, b)Sb(a, b)Sd(a, b)

Parameterized T-norm and S-norm

• Parameterized T-norms and T-conorms have been proposed by several researchers:▫ Yager▫ Schweizer and Sklar▫ Dubois and Prade

Hamacher▫ Hamacher▫ Frank▫ SugenoSugeno▫ Dombi


• Schweizer and Sklar

It is easy to check thatIt is easy to check that

• Yager (q > 0)

• Dubois and Prade ( α ε [0,1])


• Hamacher (γ > 0)

Frank (s > 0)• Frank (s > 0)

• Sugeno (λ ≥ -1)

• Dombi (λ > 0)

T-norm Operator

T-conorm or S-norm

Note that …

• Normality and convexity may be lost when we operate on f t b th t d d ti f i t ti dfuzzy sets by the standard operations of intersection and complement.

• The fuzzy intersection and fuzzy union will satisfy all the properties of the Boolean lattice listed in previous slidesproperties of the Boolean lattice listed in previous slides except the low of contradiction and the low of excluded middle (union and intersection of a set with its complement).

• The law of contradiction• The law of contradiction

• To verify that the law of contradiction is violated for fuzzy sets, we need

AA

y yonly to show that

is violated for at least one .Thi i i th ti i b i l i l t d f l

0)](1),(min[ xAxA

Xx▫ This is easy since the equation is obviously violated for any value

, and is satisfied only for )1,0()( xA }.1,0{)( xA

Dual triple• Let the triple <i, u ,c> denote that i and u are dual with respect to c • We can easily verify the following are dual triples:• We can easily verify the following are dual triples:

Aggregation operationsgg g

• Aggregation operations on fuzzy sets are operations by gg g p y p ywhich several fuzzy sets are combined in a desirable way to produce a single fuzzy set.


• Axiomatic requirements of aggregation operations:gg g

• Additional axiomatic requirement:


• Fuzzy intersections and unions are aggregation operations on fuzzy set. However fuzzy intersections and unions are not idempotent (Axiom h4) with• However, fuzzy intersections and unions are not idempotent (Axiom h4), with the exception of the standard min and max operations.

• It is significant that any aggregation operation h that satisfies Axioms h2 (and if it is symmetric) satisfies also the inequalities:y ) q

• All aggregation operations between the standard fuzzy intersection and the t d d f i id t tstandard fuzzy union are idempotent.

Aggregation operationsgg g• Generalized means:

• The lower bound:

• The upper bound:• The upper bound:

• The harmonic mean:

• The arithmetic mean:


• Another class of averaging operations that covers the entire interval between the min and max operations is called the class of orderedbetween the min and max operations is called the class of ordered weighted averaging (OWA) operations.

• For example:For example:

So, in general (Norm operator)g ( )

• Norm operations:▫ One kind of binary aggregation operations that satisfy the properties

of monotonicity, commutativity, and associativity of t-norms and t-conorms, but replace the boundary conditions of t-norms and t-conorms with weaker boundary conditions

h(0, 0) = 0 and h(1, 1) = 1.

When it becomes t-norm?

• Norm operations:▫ When a norm operation also has the property h(a, 1) = a, it

becomes a t-norm.▫ When it has also the property h(a, 0) = a, it becomes a t-conorm.▫ Otherwise, it is an associative averaging operation.▫ Norm operations cover the whole range of aggregating operations

• For example: a norm operation neither t-norm nor t-conormp p

Readingg

• J-S R Jang and C-T Sun Neuro-FuzzyJ S R Jang and C T Sun, Neuro Fuzzy and Soft Computing, Prentice Hall, 1997 (Chapter 2)(Chapter 2).

lectures 2&3: fuzzy sets

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