2 Classical and Fuzzy Sets

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					       2. Classical and Fuzzy Sets




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       3.1. Crisp vs. Fuzzy Sets:
       The concept of a set is fundamental to mathematics. How-ever,
       our own language uses sets extensively. For example, car
       indicates the set of cars. When we say a car, we mean one out of
       the set of cars.

          The classical example in fuzzy sets is tall men. The elements
       of the fuzzy set “tall persons” are all persons, but their degrees of
       membership depend on their height.




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  Example: Crisp vs. Fuzzy Sets: TALL-Men set


          Person      Height    Degree of Membership
                                  Crisp         Fuzzy
        Person 1       205          1           1.00
        Person 2       182          1           0.81
        Person 3       175          0           0.38
        Person 4       167          0           0.10
        Person 5       155          0           0.04
        Person 6       152          0           0.00


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            Crisp Set
       (clearly defined Set)

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         Unlike two-valued Boolean logic, fuzzy logic is multi-valued. It
       deals with degrees of membership and degrees of truth.

         Fuzzy logic uses the continuum of logical values between 0
       (completely false) and 1 (completely true), accepting that things
       can be partly true and partly false at the same time.




             0    0     0 1     1     1   0 0   0.2   0.4   0.6   0.8   1 1
                 (a) Boolean Logic.         (b) Multi-valued Logic.




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Example: Fuzzy set of Tall Men




   The x-axis represents the Universe of Discourse (universe of all
available information on a given problem/ the space where the fuzzy
variables are defined). According to this representation, the universe
of person’s heights consists of all ’tall’ men.
  The y-axis represents the Membership Value of the fuzzy set. In
our case, the fuzzy set of “tall men” maps height values into
corresponding membership values.
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 3.2. Crisp Sets:
 Let X be the universe of discourse and its elements be denoted as x.

    In the classical set theory, a crisp set A of X is defined by a
 function fA(x) called the characteristic function of A:
        fA(x) : X  {0/1}, where
 This set maps universe X to a set of two elements. For any element
 x of universe X, characteristic function fA(x) is equal to 1 if x is an
 element of set A, and is equal to 0 if x is not an element of A.

 x∈X            ⇒ x belongs to X
 Notations:

 A⊂B            ⇒ A is fully contained in B (if x ∈ A, then x ∈ B)
 A⊆B            ⇒ A is contained in or is equivalent to B
                ⇒ A ⊆ B and B ⊆ A (A is equivalent to B)
 The null set, ∅, is the set containing no elements
 (A ↔ B)



 Conventional or crisp sets are binary. An element either
 belongs to the set (True/1) or doesn’t (False/0) .
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3.3. Fuzzy Sets:
  In the fuzzy theory, a fuzzy set A of universe X is defined by function
µA(x) called the membership function of set A
       µA(x) : X  {0, 1}, where          µA(x) = 1 if x is totally in A;
                                          µA(x) = 0 if x is not in A;
                                          0 < µA(x) < 1 if x is partly in A.
This set allows a continuum of possible choices. For any element x of
universe X, membership function µA(x) equals the degree to which x
is an element of set A. This degree, a value between 0 and 1,
represents the degree of membership, also called membership value,
of element x in set A.

Thus
Membership functions of fuzzy sets are continuous. An element
can belong to the set to a certain degree {0  1} .

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                                  First determine the member-
3.4. Fuzzy Set Representation:    ship functions:
                                 In the “tall men” example, we
                                 can obtain Crisp set of tall,
                                 short and average men, and
                                 also of Fuzzy set.
                                   The universe of discourse –
                                  the men’s heights – consists
                                  of three sets: short, average
                                  and tall men. A man who is
                                  184 cm tall is:
                                  Crisp: a member of the tall
                                  set but not of the others
                                  Fuzzy: he is a member of
                                  average men set with a
                                  degree of membership of
                                  0.1, and at the same time,
                                  he is also a member of the
                                  tall men set with a degree
                                  of 0.4, and is not a member
                                  of short men set
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Fuzzy Set Representation (Contd.)
 There are Continuous universes
comprising of an infinite collection of
elements (see fuzzy set of Tall Men on
slide 6) , and Discrete universes, that
are composed of a finite collection of
elements. In the last case when
elements are discrete the grade of
membership can be represented by:
     A = {(1,0.7), (2, 0.5),(3,1.0)} Or
      A = 0.7/1 + 0.5/2+1.0/3
   be sure to notice that the symbol ‘+’ implies not addition but
   union.
   More Generally, we use                      Or




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  Example: Discrete vs. Continuous Membership functions




         A discrete membership function for ”x is close to 1”




       A Continuous membership function for ”x is close to 1”


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3.5. Standard Operations on Fuzzy Sets:
Consider a universal set X which is defined on the age domain.
               X = {5, 15, 25, 35, 45, 55, 65, 75, 85}

       age(element)   infant   young     adult     senior
             5          0        0         0         0
            15          0       0.2       0.1        0
            25          0        1        0,9        0
            35          0       0.8        1         0
            45          0       0.4        1        0.1
            55          0       0.1        1        0.2
            65          0        0         1        0.6
            75          0        0         1         1
            85          0        0         1         1

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  1) Complement:
  Definition:
  Crisp Sets: Who does not belong to the set?
  Fuzzy Sets: How much do elements not belong to the set?

  The complement of a set is an opposite of this set. For
  example, if we have the set of tall men, its complement is
  the set of NOT tall men. When we remove the tall men set
  from the universe of discourse, we obtain the complement.




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  The complement set    of a fuzzy set A is found as
  follows:
  The degreee of Membership is given by:



  In our example A is the “adult” set and is given by:

           A = {(5, 0), (15, 0.1), (25, 0.9),(35,1)}.
  Then:

              = {(5, 1), (15, 0.9), (25, 0.1),(35,0}.


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       Not Adult




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  2. Union ( )
  Definition
  Crisp Sets: Which element belongs to either set?
  Fuzzy Sets: How much of the element is in either set?

  The union of two crisp sets consists of every element that
     falls into either set. For example, the union of tall men
     and fat men contains all men who are tall OR fat.

  In fuzzy sets, the union is the largest membership value of
      the element in either set.




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  Membership value of member x in the union takes the greater
    value of membership between A and B



  Of course, A and B are subsets of A B, now if B represents
      “young” and is given by:
  B= {(5, 0), (15, 0.2), (25, 1),(35,0.8 ), (45, 0.4),(55,0.2)}
  and as before adult A={(5, 0), (15, 0.1), (25, 0.9),(35,1)}

  Then the union of “young” and “adult” is
   A    B = {(15,0.2), (25,1), (35,1), (45,1), (55,1),
     (65,1),(75,1), (85,1)}

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       Young   Adult




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  3. Intersection ( ( )
  Definition
  Crisp Sets: Which element belongs to both sets?
  Fuzzy Sets: How much of the element is in both sets?

  In classical set theory, an intersection between two sets
      contains the elements shared by these sets. For
      example, the intersection of the set of tall men and the
      set of fat men is the area where these sets overlap. In
      fuzzy sets, an element may partly belong to both sets
      with different memberships.

  A fuzzy intersection is the lower membership in both sets of
      each element.
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  Intersection of fuzzy sets A and B takes the smaller value of
      membership function between A and B.



  Intersection A     B is a subset of A or B. For instance, the
      intersection of “young” and “adult” is

  A    B= {(15, 0.1), (25, 0.9), (35, 0.8), (45, 0.4), (55, 0.1)}.

  Note that complement, union and intersection sets are
  applicable even if membership function is restricted to 0 or 1
  (i.e. crisp set)

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       Young   Adult




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  Another Example:
  Let A be a fuzzy interval “between 5 and 8” and B be a fuzzy
  number “about 4”.




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  The following figure shows the fuzzy set “between 5 and 8” AND
  “about 4” (Intersection)




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  The Fuzzy set “between 5 and 8” OR “about 4” is shown in the
  next figure (Union)




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  This figure gives an example for a negation. The blue line is the
  NEGATION (Complement) of the fuzzy set A. i.e. “not a fuzzy
  interval between 5 and 8”




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Example:
Consider the fuzzy set of the points
included in a circle




                                            Distance membership function




                                   Representation of the grade of
                                   membership of the inclusion within the
                                   circle with black representing the
                                   points not in the set i.e. μP1=0

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  2010                        Hany Selim                             27
                             Representation of the grade of
                             membership of the inclusion within
                             a shifted circle with black
                             representing the points not in the
                             set i.e. μP2




       Union of the 2 sets
           μP1 μP2


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                              The intersection of the two sets
                                       μP1     μP2




The complement of the first
set above:
         μP1


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                                     The intersection of the first set
                                     and its complement is not empty
                                     (remember by Crisp A.A=0)




The union of the first set and its
complement is not 1 (remember
by Crisp A+A=1)



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  3.6. Properties of Fuzzy Sets:
       Equality of two fuzzy sets

       Inclusion of one set into another fuzzy set

       Cardinality of a fuzzy set

       An empty fuzzy set

       -cuts (alpha-cuts)

       Fuzzy set Normality

       Height of the Fuzzy set

       Fuzzy set Core

       Fuzzy set Support

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  1. Equality:
  Fuzzy set A is considered equal to a fuzzy set B, IF AND
  ONLY IF (iff):

                A(x) = B(x), xX

         A = 0.3/1 + 0.5/2 + 1/3
         B = 0.3/1 + 0.5/2 + 1/3

         therefore A = B




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  2. Inclusion:
  Inclusion of one fuzzy set into another fuzzy set. Fuzzy set
  A  X is included in (is a subset of) another fuzzy set,
  B  X:
                 A(x)  B(x), xX

         Consider X = {1, 2, 3} and sets A and B

         A = 0.3/1 + 0.5/2 + 1/3;
         B = 0.5/1 + 0.55/2 + 1/3

         then A is a subset of B, or A  B




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  3. Cardinality:
  Cardinality of a non-fuzzy set, Z, is the number of elements
  in Z. BUT the cardinality of a fuzzy set A, the so-called
  SIGMA COUNT, is expressed as a SUM of the values of the
  membership function of A, A(x):

         cardA = A(x1) + A(x2) + … A(xn) = ΣA(xi),
  for i=1..n

  Consider X = {1, 2, 3} and sets A and B

          A = 0.3/1 + 0.5/2 + 1/3;
          B = 0.5/1 + 0.55/2 + 1/3
   then
          cardA = 1.8
          cardB = 2.05
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  4. Empty Fuzzy Set:
  A fuzzy set A is empty, IF AND ONLY IF:

               A(x) = 0, xX

        Consider X = {1, 2, 3} and set A

        A = 0/1 + 0/2 + 0/3

        then A is empty




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  5. Alpha-cut:
  An -cut or -level set of a fuzzy set A  X is a Crisp set
  A  X, such that:

                  A={A(x), xX}

         Consider X = {15,25,35,45,55} representing age
  and set A representing young
          A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55

          then     A0.5 = {25, 35},
                   A0.1 = {15, 25, 35, 45},
                   A1 = {25}
  A0.5 means “the age that we can say of being young with possibility
  not less than 0.5”.

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  6. Fuzzy set Normality:
  A fuzzy subset of X is called normal if there exists at least

            one element xX such that A(x) = 1

  • A fuzzy subset that is not normal is called subnormal.

  • All crisp subsets except for the null set are normal.

  •Example:
  A = 0.2/15 + 1/25 + 0.8/35+0.3/45+0/55
  is a Normal set.
  But
  A = 0.2/15 + 0.9/25 + 0.8/35+0.3/45+0/55
  Is a Subnormal set.
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  7. Height of the Fuzzy set:
  The height of a fuzzy subset A is the large membership
  grade of an element in A

                 height(A) = maxx (A(x))

  Example:
  If    A = 0.2/15 + 0.6/25 + 0.8/35+0.3/45+0/55
  then  height(A)=0.8




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  8. Fuzzy set Core:
  the core of A is the crisp subset of X consisting of all
  elements with membership grade = 1:

              core(A) = {x A(x) = 1 and xX}

  Example:
  If    A = 0.2/15 + 1/25 + 0.8/35+1/45+0/55
  then  core(A)={25,45}




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  9. Fuzzy set Support:
  the support of A is the crisp subset of X consisting of all
  elements with a degree of membership differing from zero:

                supp(A) = {x A(x)  0 and xX}

  Example:
  If    A = 0.2/15 + 0/25 + 0.8/35+1/45+0/55
  then  supp(A)={15, 35,45}




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  3.7. Fuzzy Set Math Operations:

                      kA = {kA(x), xX}
  Let k =0.5, and
                A = {0.5/a, 0.3/b, 0.2/c, 1/d}
  then
                kA = {0.25/a, 0.15/b, 0.1/c, 0.5/d}

                      Ak = {A(x)k, xX}
  Let k =2, and
                  A = {0.5/a, 0.3/b, 0.2/c, 1/d}
  then
                  Ak = {0.25/a, 0.09/b, 0.04/c, 1/d}


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  3.8. Properties of Fuzzy:




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  Quiz:

  Consider two fuzzy subsets of the set X,
                  X = {a, b, c, d, e }

          referred to as A and B

                  A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
          and
                  B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}




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  1. Cardinality:
  card(A) =                 A = {1/a, 0.3/b, 0.3/c ,0.8/d, 0/e}
                            B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}

  card(B) =

  2. Complement of A and of B
  A=

  B=


  3. Union:
  AB=



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  4. Intersection:   A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
  AB=               B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e}



  5. a-cut:
  A0.2 =

  A0.3 =

  A0.8 =

  A1 =




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  6. Support:
  supp(A) =
                       A = {1/a, 0.3/b, 0.3/c, 0.8/d, 0/e}
                       B = {0.6/a, 0.9/b, 0.1/c, 0.3/d, 0.2/e
  supp(B) =


  7. Core:
  core(A) =

  core(B) =


  8. aA    for a=0.5


  9. Aa:   for a=2

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  Example:




  Where the possibility for X=3 is 1, the probability is only 0.1

  The example shows, that a possible event does not imply that it is
  probable. However, if it is probable it must also be possible. You
  might view a fuzzy membership function as your personal
  distribution, in contrast with a statistical distribution based on
  observations

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