This document discusses fuzzy sets and their applications. It defines key terms like membership functions and fuzzy sets. A fuzzy set allows for partial membership where elements have degrees of membership between 0 and 1, unlike crisp sets which have binary membership. Eigen fuzzy sets are discussed as fuzzy sets that remain unchanged after applying a max-min composition with a given fuzzy relation. An example is provided to illustrate how eigen fuzzy set theory can be used to find the maximum and minimum satisfaction levels of mobile users for different features. The results can help measure customer satisfaction levels and identify areas for improvement.

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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