The Mathematics of Fuzzy Sets Part 1

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The Mathematics of Fuzzy SetsPart 1

ECE 5831 - Fall 2006

• Prof. Marian S. Stachowicz
• Laboratory for Intelligent Systems
• ECE Department, University of Minnesota Duluth
• September 11, 2006

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Outline

• TYPES OF UNCERTAINTY
• Fuzzy Sets and Basic Operations on Fuzzy Sets
• Further Operations on Fuzzy Sets

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References for reading

• 1. Timothy Ross,
• Fuzzy Logic with Engineering Application, John
  Wiley Sons, Ltd, 2004
• Chapter 1 and 2
• 2. M.S. Stachowicz, Lance Beall, Fuzzy Logic
  Package,
• Version-2 for Mathematica 5.1, Wolfram
  Research, Inc., 2003
• - Demonstration Notebook 2.1, 2.2, 2.7.1,
  2.7.2, 2.8.1, 2.8.2, 2.8.3, 2.8.5, 2.8.6, 2.8.8,
  2.8.9
• - Manual 1.1, 1.3.1, 1.3.2, 1.3.4, 1.4, 1.5
• 3. G.J. Klir, Bo Yuan, Fuzzy Set and Fuzzy Logic,
  Prentice Hall, 1995
• Chapters 1, 2, and 3

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Randomness versus Fuzziness

• Randomness refers to an event that my or may not
  occur.
• Randomness frequency of car accidents.
• Fuzziness refers the boundary of a set that is
  not precise.
• Fuzziness seriousness of a car
  accident. Prof. George J.Klir
  
  PROFESSOR
  GEJ.

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TYPES OF UNCERTAINTY

• STOCHASTIC UNCERTAINTY
• THE PROBABILITY OF HITTING THE TARGET IS 0.8.
• LEXICAL UNCERTAINTY
• WE WILL PROBABLY HAVE A SUCCESFUL FINANCIAL YEAR.

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FUZZY SETS THEORY versus PROBABILITY THEORY

• Patients suffering from hepatitis show in
• 60 of all cases high fever, in 45 of all
  cases a yellowish colored skin, and in 30 of all
  cases nausea.

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What are Fuzzy Sets?
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Fuzzy setLotfi A. Zadeh1965

• A fuzzy subset A of a universe of discourse U
• is characterized by a membership function
• ?A U ? 0,1
• which associates with each element u of U a
  number ?A(u) in the interval 0,1, which ?A(u)
  representing the grade of membership of u in A.

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Fuzzy Sets

• Fuzzy set A defined in the universal space U is
  a function defined in U which assumes values in
  the range 0,1 .
• A U ? 0, 1

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Problem 1 Given the set U 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, describe the set of
prime numbers.
A u in U u is a prime number
The elements of the set are defined unequivocally
as A 2, 3, 5, 7, 11
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Problem 2 Suppose we want to
describe the set of small numbers using the same
set U.
M u in U u is a small number
Now, it is not so easy to define the set. We can
use a sharp transition like the following,
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An alternative way to define the set would be to
use a smooth transition.
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L.A. Zadeh, Fuzzy sets, Information and Control,
vol. 8, pp. 338-353, 1965
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Characteristic Function

A U ? 0, 1

Membership Function
M U ? 0, 1
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Universal Set

• U - is the universe of discourse, or universal
  set, which contains all the possible elements of
  concern in each particular context of
  applications.

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Membership function

• The membership function M maps each element of U
  to a membership grade
• ( or membership value) between 0 and 1.

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Presentation of a fuzzy set

• A fuzzy set M, in the universal set can be
  presented by
• - list form,
• - rule form,
• - membership function form.

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List form of a fuzzy set

• M 1,1,2,1,3,0.9,4,0.7,5,0.3,6,0.1
  ,
• 7,0,8,0,9,0,10,0,11,0,12,0,
• where M is the membership function (MF) for
  fuzzy set M.
• Note
• The list form can be used only for finite sets.

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Fuzzy Logic Package form M.S. Stachowicz
Lance Beall ,1995 2003

• M1,1,2,1,3,0.9,4,0.7,5,0.3,6,0
  .1
• and
• U1,2,3,4,5,6,7,8,9,10,11,12

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Rule form of a fuzzy set

• M u ? U?u meets some conditions,
• where symbol ?denotes the phrase such as.

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Membership form of a fuzzy set

Let A1 be a fuzzy set named numbers closed to
zero A1(u) exp(-u2) for u ? -3,3 A1(0)1
A1(2)exp(-4) A1(-2)exp(-4)
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Numbers closed to zero
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Representation of a fuzzy set

A fuzzy set A in U may be represented as a set
of ordered pairs of generic element u and its
membership value A(u),

A u, A(u) ? u ? U
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Representation of a fuzzy set

When U is continuous, A is commonly written
as ?u A(u) ?u where integral sign does not
denote integration it denotes the collection of
all points u ? U with the associated MF A(u).

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Representation of a fuzzy set

When U is discrete, A is commonly written
as ? A(u) ?u, where the summation sign
does not represent arithmetic addition.

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Example 1

• Let U be the integer from 1 to 10, that
• U 1,2,,9,10. The fuzzy set Severalmay be
  defined as using
• - the summation notation
• Several0.5/3 0.8/4 1/5 1/6 0.8/7
  0.5/8
• - FLP notation
• Several3.0.5,4,0.8,5,1,6,1,7,0.8,8,0.
  5

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Fuzzy Sets

• Analog fuzzy sets
• A U ? 0, 1
• Discrete fuzzy sets
• A u1,u2,u3,...,us ? 0, 1
• Digital fuzzy sets
• A u1,u2,u3,...,us ?0, 1/n-1, 2/n-1 ,
  3/n-1,..., n-2/n- 1, 1

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Analog fuzzy sets

• If a continuous-universal space membership
  function A can take on any value (the grade of
  membership) in the continuous interval 0, 1.
• A U ? 0, 1

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Example 1 Analog fuzzy sets

• Fuzzy set with continuous U. Let U R be the
  set of possible ages for teens. Then the fuzzy
  set A " about 10 year old" may be
  expressed.

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Discrete fuzzy setsM.S.Stachowicz M.
Kochanska, 1982

• A discrete-universal space membership function
  has a value (the grade of membership) only at
  discrete points in universal space
• A u1,u2,u3,...,us ? 0, 1

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Example 2 Discrete fuzzy sets
Let U 0, 20, 0.5 then the discrete fuzzy
set A has a form
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Digital fuzzy sets M.S.Stachowicz M.
Kochanska, 1982

• If a discrete-universal membership function can
  take only a finite number n ? 2 of distinct
  values
• A u1,u2,...,us ? 0, 1/n-1, 2/n-1 ,...,
  n-2/n- 1, 1

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Example 3 Digital fuzzy sets

• A digital fuzzy set A with discrete universal
  space and n 8 levels
• has a form

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Digital Fuzzy Sets
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Digital Fuzzy Sets

• DFS can be viewed and manipulated in the same
  manner as the infinite valued fuzzy sets.
• Let compare the discrete and digital form of our
  fuzzy sets using the concept of the Hamming
  distance
• 4., 1.8667, 1.33333, 0.933333, 0.8
• As you might expect, as n goes to infinity,
  digital fuzzy set becomes discrete fuzzy set.

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Basic concepts and terminology

• The concepts of support, fuzzy singleton,
  crossover point, height, normal FS,
• ?-cut, and convex fuzzy set are defined as
  follows

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The support of fuzzy set A

• The support of a fuzzy set A in the universal set
  U is a crisp set that contains all the elements
  of U that have nonzero membership values in A,
  that is,
• supp(A) u ? U A(u) gt 0

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Fuzzy singleton

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

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Crossover point

• The crossover point of a fuzzy set is the point
  in U whose membership value in A equals 0.5.

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Height

• The height of a fuzzy set is the largest
  membership value attained by any point.
• If the height of fuzzy set equals one, it is
  called a normal fuzzy set.

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An ?-cut of a fuzzy set

• An ?-cut of a fuzzy set A is a crisp set ?A
• that contains all the elements in U that have
  membership value in A greater than or equal to ?.
• Crisp set ?A xA(x) ? ?

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Convex fuzzy set

• A fuzzy set A is convex if and only if its
• ? -cuts ?A is a convex set for any ? in the
  interval ? ? (0,1.
• A? x1 (1- ? ) x2 ? minA(x1),A(x2)
• for all x1, x2 ? Rn and all ? ? 0,1.

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Operations on fuzzy sets

• Inclusion
• Equality
• Standard Complement
• Standard Union
• Standard Intersection

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Inclusion

• Let X and Y be fuzzy sets defined in the same
  universal space U.
• We say that the fuzzy set X is included in the
  fuzzy set Y if and only if
• for every u in the set U we have X(u) Y(u)

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Equality

• Let X and Y be fuzzy sets defined in the same
  universal space U.
• We say that sets X and Y are equal, which is
  denoted X Y if and only if for all u in the set
  U, X(u) Y(u).

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Standard complement

• Let X be fuzzy sets defined in the universal
  space U.
• We say that the fuzzy set Y is a complement of
  the fuzzy set X, if and only if, for all u in the
  set U,
• Y(u) 1 - X(u).

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Standard union

• Let X and Y be fuzzy sets defined in the space U.
  We define the union of those sets as the smallest
  (in the sense of the inclusion) fuzzy set that
  contains both X and Y.
• ? u? U, (X ? Y)(u) Max(X(u), Y(u)).

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Standard union

• ? u ? U, (X ? Y)(u) Max(X(u), Y(u)).

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Standard intersection

• Let X and Y be fuzzy sets defined in the space U.
  We define the intersection of those sets as the
  greatest (in the sense of the inclusion) fuzzy
  set that included both in X and Y.
• ? u?U, (X ? Y)(u) Min(X(u), Y(u)).

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Standard intersection

• ? u?U, (X ? Y)(u) Min(X(u), Y(u)).

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Properties of crisp set operations

• Involution (A) A
• Commutativity A U B B U A
  A ? B B ? A
• Associativity (A U B) U C A U (B U C)
• (A ? B) ? C A ? (B ? C)
• Distributivity A ? (B U C) (A ? B) U (A ? C)
• A U (B ? C) (A U B) ? (A U C)
• Idempotents A U A A
• A ? A A

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Properties of fuzzy set operations

• Involution (A) A
• Commutativity A U B B U A
  A ? B B ? A
• Associativity (A U B) U C A U (B U C)
• (A ? B) ? C A ? (B ? C)
• Distributivity A ? (B U C) (A ? B) U (A ? C)
• A U (B ? C) (A U B) ? (A U C)
• Idempotents A U A A
• A ? A A

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Properties of crisp set operations

• Absorption A U (A ? B) A
• A ? (A U B) A
• Absorption by X and ? A U X X
• A ? ? ?
• Identity A U ? A
• A ? X A
• De Morgans laws (A ? B) A U B
• (A U B) A ? B

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Properties of fuzzy set operations

• Absorption A U (A ? B) A
• A ? (A U B) A
• Absorption by X and ? A U X X
• A ? ? ?
• Identity A U ? A
• A ? X A
• De Morgans laws (A ? B) A U B
• (A U B) A ? B

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Properties of crisp set operations

• Law of contradiction A ? A ?
• Law of excluded middle A U A X

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Properties of fuzzy set operations

• Law of contradiction A ? A ? ?
• Law of excluded middle A U A? X

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Law of excluded middle A U A? X

• FuzzyPlotMEDIUM U ComplementMEDIUM

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Law of contradiction A ? A ? ?

• FuzzyPlotMEDIUM ? ComplementMEDIUM

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Standard operators 1

• When the range of grade of membership is
  restricted to the set 0,1, these functions
  perform like the corresponding operators
• for Cantor's sets.

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Standard operators 2

• If any error e is associated with the grade of
  membership A(u) and B(u), then the maximum error
  associated with the grade of membership of u in
  A', UnionA, B, and IntersectionA, B remains e.

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Characteristic function

• Ch. de la Valle Poussin 1950, Integrales de
  Lebesque, fonction d'ensemble, classes de Baire,
  2-e ed., Paris, Gauthier-Villars.

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Membership function

• L. A. Zadeh 1965, Fuzzy sets,
• Information and Control, volume 8,
• pp. 338-353.

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• Goguen, J.A.1967 L-fuzzy sets,
• J. of Math Analysis and Applications,
• 18(1), pp.145-174.

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• M. S. Stachowicz and M. E. Kochanska 1982,
  Graphic interpretation of fuzzy sets and fuzzy
  relations, Mathematics at the Service of Man.
  Edited by A. Ballester, D. Cardus, and E.
  Trillas, based on materials of Second World
  Conference, Universidad Politecnica Las Palmas,
  Spain.

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www.wolfram.com/fuzzylogic
M.S. STACHOWICZ and L. BEALL 1995, 2003
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Individual decision making
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Individual decision making

• A decision is characterized by components
• Universal space U of possible actions
• a set of goals Gi (i ? Nn) defined on U
• a set constraints Cj (j ? Nn) defined on U.
• Decision is determined by an aggregation operator.

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Example Job Selection

• Suppose that John from UMD needs to decide which
  of the four possible jobs, say
• g(a1) 40,000
• g(a2) 45,000
• g(a3) 50,000
• g(a4) 60,000
• to choose.

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The constraints

• His goal is to choose a job that offers a high
  salary under the constraints that the job is
  interesting and within close driving distance.

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• In this case, the goal and constraints are all
  uncertain concepts and we need to use fuzzy sets
  to represent these concepts.

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Goal High salary-indirect form

• CF FuzzyTrapezoid37, 64, 75, 75,
• UniversalSpace -gt 0, 80, 5
• FuzzySet 40, 0.11, 45, 0.3, 50, 0.48,
  55, 0.67, 60, 0.85, 65, 1, 70, 1, 75, 1

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Fuzzy goal G High salary-direct form

• G FuzzySet1, .11, 2, .3, 3, .48, 4,
  .85

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Constraint C1 Interesting job
C1 FuzzySet1, .4, 2, .6, 3, .2, 4,
.2
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Constraint C2 Close driving
C2 FuzzySet1, .1, 2, .9, 3, .7, 4, 1
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Concept of desirable job

• D IntersectionG, C1, C2
• FuzzySet1, 0.1, 2, 0.3, 3, 0.2, 4,
  0.2

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For John from UMD 45,000.

• The final result from the above analysis is a2,
  which is the most desirable job among the four
  available jobs under the given goal G and
  constraints C1 and C2.
• g(a2) 45,000 with D(a2) 0.3

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Example Optimal dividends

• The board of directors of a company needs to
  determine the optimal dividend to paid to the
  shareholders. For financial reasons, the dividend
  should be attractive (goal G) for reasons of
  wage negotiations, it should be modest
  (constraint C).
• The U is a set of possible dividends actions.

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• In this case, the goal and constraint are both
  uncertain concepts and we need to use fuzzy sets
  to represent these concepts.

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Example Optimal dividends
C modest
G attractive
?
û optimal dividends
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Optimal dividends

• OptimalDividends CoreNormalizeIntersection
• modest, attractive

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Comments

• Since this method ignores information concerning
  any of the other alternatives, it may not be
  desirable in all situations.
• An averaging operator may be used to reflect a
  some degree of positive compensation exists among
  goals and constrains.
• When U is defined on R, it is preferable to
  determine û by appropriate defuzzification method.

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Questions ?
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Have a nice evening.