Crisp Set vs Fuzzy Set, Eigen Fuzzy Set, Applications of Fuzzy Set

1. S.SWATHI SUNDARI, M.Sc.,M.Phil.,
Assistant Professor of Mathematics
V.V.Vanniaperumal College for Women
Virudhunagar
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Fuzzy Sets
And Applications

2. Fuzzy Sets
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3. Basic Terminology
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 A membership function:
 A characteristic function: the values assigned to the
elements of the universal set fall within a specified
range and indicate the membership grade of these
elements in the set.
 Larger values denote higher degrees of set
membership.
 Fuzzy Set:
The membership function of a fuzzy set A is denoted
by :
]1,0[: XA
A

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 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].
 The universal set X is always a crisp set.
Crisp set vs. Fuzzy
set
A traditional crisp set A fuzzy set

5. Crisp Set Vs. Fuzzy Set
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 The crisp set is defined in such a way as to partition the
individuals in some given universe of discourse into two
groups: members and nonmembers.
However, many classification concepts do not exhibit
this characteristic.
For example, the set of tall people, expensive cars, or
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 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 of
30%, and 0 to a cloud cover of 75%.

6. Eigen Fuzzy Sets
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Let R be a fuzzy relation between the
elements of a finite set X and let A be a fuzzy subset
of X. The Max-Min Composition of R and A gives B,
(AOR=B, B X).When B becomes equal to A, we say
A is an Eigen fuzzy set associated with given relation
R.


7. Applications of Eigen Fuzzy Sets
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Mobile Users Satisfaction:
By applying Eigen Fuzzy set theory we
can find maximum and minimum satisfaction level
of (Particular model) mobile customers or users.

8. For Example…
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Choose Nokia Express Music 5800 and asked from
10 users about different feature of this model.
 = User Friendly or Usability
 = Touch Screen Response
 = Battery Timing

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 Let we denote the set of features by F as
 The satisfaction of the jth feature is equal or stronger
than kth feature in a customer j,k=1,2,…n.
 Each pair of relation ( , ) has a
membership degree from range [0,1], indicated the
grade to which the statement defining is true
for jth and kth feature.
 We use formula,
Where j,k= 1,2,…n

10. Sign figurations for
the three given
features
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11. In an above example…
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 Where m is total number of users in this case, m=10.
 Where b stands for our users asked 3 different features
of Nokia express music 5800.
 Suppose “-” assigned for low satisfaction and “+”
assigned for good or greater satisfactions to count b.
 Then we have to find the fuzzy relation by using
the membership degree of pair from the table.

12. To find
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 Formula :
We find a diagonal entries of the above matrix, we get,

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 Putting all the values, and we get,
 Here A2=A3, so A3 is accepted as the GEFS.

14. To find
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 In this way, we can find a least Eigen fuzzy set as
follows:
 A3 is least Eigen fuzzy set.

15. Conclusion:
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 By interpreting the degrees of and in
percentage scale, we can say that the customer’s
satisfaction level of Nokia express music 5800 is
about 50% - 60% and 40% - 65% and in min
satisfaction 50% and max 65%.
 We can verify that

16. The result shows…
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 How the highest customer satisfaction level can be
measured by applying EFST.
 How customer satisfaction can be increased for any
product, and which attributes of any product quality
needs to improve.
 In future, EFST can be used for image enhancement
techniques to improve perceptual image quality and
speech enhancement and quality.
 EFST used for improving the quality of any product
based on the statistical data.

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